Guinan, perhaps her whole species, is able to understand the 'correct' state of the universe. Take for example in Yesterday's Enterprise ^[1] she knows details about the 'other' universe. The universe as it was during the rest of the series with the Enterprise as a ship of peace, not a ship of war.

This is, essentially, the counter power to Q's ability to alter the universe at will. It suggests that the power the Q hold is transparent to Guinan. She can see through their alterations to the mechanics of those actions and the state of the universe before those actions. Possibly, she can counter their magical technology, although that's just conjecture.

Just being able to understand the 'true' state of things would be enough to limit Q's power, which would be frightening to him. And what the Q do to the universe would be equally as upsetting to Guinan.

When Guinan was ripped from the Nexus in "Generations", part of her was still left there. She told Picard she couldn't go back with him because she was already there. The part of her left in the Nexus, a place where time had no meaning, gave her her sixth sense. This explanation was in the original Generations script but was later edited.

GUINAN : It took a long time, but eventually I learned to live with it. And I began to realize that my experience in the Nexus had changed me...I knew things about people...about events...about time...

PICARD : Your "sixth sense"... I've always wondered where it came from

I have always wondered about this myself. I would say that while her species does have incredible mental abilities, they would not be anywhere near a match for a Q. The Borg ran roughshod over her people. We know the Borg are not at the level of the Q. Here are some theories:

1. Guinan was at the point where she she was going to possibly attack him with her mind, regardless of her ability to damage him. We have seen Worf ready to attack Q despite his inability, and we have seen Sisko actually punch Q. So she may have been "blowing smoke" or angry enough to attack with out causing damage.

2. I am much more intrigued by this possibility: What if she possesed a Q weapon, unseen by the crew but deadly to Q? We know Janeway and Tuvock had Q weapons and at one point were an extreme danger to the Q. Janeway's weapon was represented by a flintrock rifle; Guinan's may have been represented with a magic spell. Q did back off. Why would she have it? Any number of Qs could have given it to her at any time over the course of her long life. Why? There are many reasons possible. Law enforcement of renegades or she like Janeway participated in a Q war, or it was given to her by another, friendlier, Q to protect her from this Q or others.

3. One last theory, and I imagine this one is a little unlikely. She may have been a Douwd or an Organian masquerading as an El-Aurian. We know both these highly advanced species, as well as the Q themselves, will do this from time to time and, if governed by a non-interference directive, would not use her abilities to help or hurt the aliens she is currently with. But she might use her abilities in self-defence against Q. He would of course see through her facade.

In Haitian Culture, a Guinan (Guinen) is a being whose mission is to bring light and aid those under his/her protection. They are perceived as elders because they may have incarnated in an ancestor. Through rituals or telepathic communication, it is possible to communicate with them. I think they are souls who have realized their higher self and chose to help mankind. Being close to the Light, they are able to warn off evil, heal and help. The path of the Guinan is to help mankind evolve, this implies that solutions come from within one's self as opposed to from some bargain with an exterior force such as Q.

I thought it was rather telling that 'Q' said she was NOT what she appeared to be, and he called her an "Imp".

I think this makes for an AWESOME back story. Originating from Germanic folklore, the imp was a lesser demon. It should also be noted that demons in Germanic legends were NOT necessarily evil. Imps were often mischievous rather than evil or harmful, and in some regions, they were portrayed as attendants of the gods.

Imps are often portrayed much the same as fairies, and in some cultures, they were considered the same beings, both sharing the same sense of free spirit and enjoyment of all things fun. Note that fairies and imps are said to be possessed of all manner of supernatural powers.

It was much later in history that people began to associate fairies with being good and imps with being malicious and evil. Though imps are often thought of as being immortal, many cultures believed that they could be damaged or harmed by certain weapons and enchantments.

ALL of this could have led to some eventual revelation that beings like 'Q', 'imps', and 'faeries' are NOT just ancient earth lore, but stories passed down about alien beings that actually visited earth from time to time.

It would have been AWESOME to see some kind of showdown between 'Q' and Guinan.

I dont think Guinan has any powers per se. However, as she does possess a superior sixth sense in understanding the universe. This is supported by her actions and Datas comment on her entire race. In regards to the Q, as previous commentator remarked, I suspected her "power" against the Q is that she is able to defend herself from his actions via neutralization. She seemed to be defending against an attempting attack by him in "Que Who." However, it is interesting that he remarked she was "not what she appeared to be." This could be his fear of her race talking, the fact that a part of her dwells in the Nexus still, or that his is truly a masquerading alien. Concerning the Nexus, I believe that experience only added to her senses. The one question that remains is what was she doing on Earth in the early 19th century?...

Guinan is an El-Aurian, which according to Star Trek canon is a race of highly intelligent, Extremely long lived space-travelers who pre-date Earth's first contact with the Vulcan. This contact was just unbeknownst to mankind at the time, as they visited in secret.

All El-Aurians are said to have latent empathic abilities of a non-descript type, being widely known as a race of "listeners", but not seeming to have the direct mind or emotion reading skill of a Betazoid. Admittedly, that last part is conjecture, because we never see or "hear" a an El-Aurian use telepathy in the show proper... however, Guinan in particular has always seemed capable of telling a troubled crewmate just what they "needed to hear" in order to find the correct answer or at least "feel" better within a given situation. Whether that is the result of some latent mental powers, or just her years of work as a bartender, one cannot be sure... but it's not impossible that is an expression of her species' latent ability.

Quote the Memory Alpha wiki:

El-Aurians prided themselves on being a race of listeners, and appeared to have a form of limited empathic ability. Some used this to help others, acting as advisers or confidants. A few turned their talent to more dubious pursuits, becoming con artists and tricksters, such as Martus Mazur, while Dr. Soran used his abilities to help bring his genocidal plan to re-enter the Nexus to fruition. (DS9: "Rivals"; Star Trek Generations) El-Aurians usually didn't experience difficulties with parents relating to their children, as El-Aurian children were usually good at listening to their parents.

El-Aurians have had encounters with numerous other alien species, including the Q and the Borg. They were the first race which disproved the Borg's logical fallacy of "resistance is futile"; even though their planet was taken over, the El-Aurians as a species escaped, survived and thrived. This longevity and the resultant encounters would likely mean they have found a few counter measures for dealing with other species as threats. Considering the power of the Q and the Borg, while outright "defeating" such opponents may not be a strong possibility, a long-lived practitioner like Guinan may well have refined and prepared plans for such tricky situations. Whether those would be of a technological, extra-normal or superhuman nature is up for debate.

Speaking of Guinan directly, she and "our" Q seemed to have some kind of ambiguous 'personal history', hence him referring to her specifically as an "imp" and the fairly obvious animosity between the two. To my knowledge, this was never elaborated on in the show itself, and with the life-span of both, it could stem from one thing, or multiple dealings. As to how Guinan could have "countered" Q, it may have something to do with her natural empathic powers, or due to something specific to her as TruTrekkie postulated due to her time in the Nexus. If that were true, though, there should have been some type of "perception altering" for Picard as well, which we never see happen.

However, we have seen that there are Q weapons which could harm these beings, so its not impossible that Q knows Guinan has access to one such weapon; we've already seen that, for all her sweet temperament, empathy and understanding, Guinan is not above "packing heat" and pulling it out when necessary. So it's not improbable that she may have access to something which could be seen as harmful to Q's kind.....

Interestingly enough, Guinan is one of the beings on the ship whom Jean Luc Picard trusts implicitly with his life, and Q warned him "She is not what she seems." This could be indicative of three things:

1. Q was trying to sow some dissention by throwing suspicion on Guinan, as a way of separating her from the Captain so he could have greater influence on Jean Luc.
2. Guinan does possess a capacity for threat [mostly to him] simply by what she knows or could know, and doesn't want a consolidation of power with her and Picard.
3. Plain, vindictive jealousy; Q has always shown a distinct interest in Picard, and this disliked El-Aurian having Picard's trust could have just been a thorn in his side, and that was a purely emotion-based jab.

While I don't think just "knowing the true nature of things" would be enough to "limit Q's power" as @DampeS8N said, I do think it could make his trickery more difficult if she possessed a limited ability to "see through" his alterations. And being so thwarted would be a minor annoyance to a god-being, one would think. However, if Guinana had any sort of power that could match or counter Q in a substantial way, it's reasonable to think she would have used such to aid the ship in other crisis situations, as opposed to sitting back and letting things unfold or resorting to a gun.

Then again, on the ship, she was dealing with mortals; Q was a different story. If that were the case, though, then the Q would have been no serious threat to the El Aruians, let alone them having to evacuate their planet to flee the Borg. It is possible that in all her years, Guinan specifically has learned some type of mental or extra-dimensional discipline [sort of like a "magic spell"] which could either prevent Q's power from directly influencing her, or even do some minor damage, but again, there is no solid evidence thereof.

Aside from the broadly expressed empathic ability and longevity, though, there is little to canonical support either Guinan herself, or her entire species having any other type of special abilities.

I have just watched the episode where Guinan was living on 18th Century Earth, so we can certainly assume she has the gift of longevity... The fact that she was also in the Nexus who suggest that she can shift between dimensions... Beyond that she wouldnt appear to have any other particular gifts.. In retrospective It would have been interesting if the Guinan character was fleshed out a bit more as we were left with so many unknowns with the guinan character.

I'm just speculating here, and making no assertions: perhaps Guinan's power over Q lies in her ability to mess with his mind. In other words, if Guinan can say just the right thing that sticks with someone and makes them feel better, she can also do the opposite. Q may have been afraid of her ability to reveal his true self. Q may be invulnerable to physical attack but not a psychological attack. Also, perhaps since Guinan can pick up on emotions, she can also project them. If a part of her is in the nexus, then she could maybe tap into the nexus' power some way.

Personally, I like that Guinan is mysterious and you never really know the big picture with her. Its part of what makes her more intruiging.

Perhaps she was given powers by the "almost respectable" Q she mentions? This means she's met more than one Q, and perhaps that other Q imbued her with special abilities (or saw to it that she got them via the Nexus).

I am watching the episode now, Q Who?

Something no one seems to mention is when Q says: "And if necessary, although I can't imagine why, I will renounce my powers and become as weak and incompetent as all of you." He looks at Guinan when he says this and she huffy and walks near Picard looking back and forth.

I think she either was a Q, was given Q abilities, or had similar powers and gave them up.

My theory is that Guinan is something similar to the TNG episode "The Survivors" where a super alien plays "human" and has a wife even - and did nothing to stop the humans he lived with from being wiped out. Ironically their encounter occurred when Q offered to join the crew - perhaps Guinan secretly did basically the same thing - but just to experience the mortal life. The Nexus theory has validity too as far as her sensitivity to time shifts but not for dealing with Q in my opinion. (I hate TNG movies.) I'd take Q at his word - she's an "imp" and probably something powerful enough to not be impressed with Q but dedicated enough to not stop pretending to be mortal even during a Borg invasion - again like "The Survivors" alien. Picard was polite enough to not react to Q outing her. Just a theory.