Define your own environment for this; here I realize it as a two column alignment; the first column is typeset in math mode, the second one in text mode; the = is added automatically, with the correct spacing.

\documentclass{article}
\usepackage{array}

\newenvironment{conditions}
  {\par\vspace{\abovedisplayskip}\noindent\begin{tabular}{>{$}l<{$} @{${}={}$} l}}
  {\end{tabular}\par\vspace{\belowdisplayskip}}

\begin{document}
An equation just to start
\begin{equation}
P+N=S_{d}
\end{equation}
where:
\begin{conditions}
 P     &  notional permeability factor \\
 N     &  number of waves \\   
 S_{d} &  damage level
\end{conditions}

\end{document}

If your conditions are overlong, then you can use a different environment, that I call conditions*, based on tabularx:

\documentclass{article}
\usepackage{tabularx}

\newenvironment{conditions*}
  {\par\vspace{\abovedisplayskip}\noindent
   \tabularx{\columnwidth}{>{$}l<{$} @{${}={}$} >{\raggedright\arraybackslash}X}}
  {\endtabularx\par\vspace{\belowdisplayskip}}

\begin{document}
An equation just to start
\begin{equation}
P+N=S_{d}
\end{equation}
where:
\begin{conditions*}
 P    &  notional permeability factor and something 
          longer that needs to be taken at the next line\\
N     &  number of waves \\
S_{d} &  damage level
\end{conditions*}

\end{document}

Variant for variable symbols

If different symbols are needed instead of = in each line, here's how.

\documentclass{article}
\usepackage{array,tabularx}

\newenvironment{conditions}
  {\par\vspace{\abovedisplayskip}\noindent
   \begin{tabular}{>{$}l<{$} @{} >{${}}c<{{}$} @{} l}}
  {\end{tabular}\par\vspace{\belowdisplayskip}}

\newenvironment{conditions*}
  {\par\vspace{\abovedisplayskip}\noindent
   \tabularx{\columnwidth}{>{$}l<{$} @{}>{${}}c<{{}$}@{} >{\raggedright\arraybackslash}X}}
  {\endtabularx\par\vspace{\belowdisplayskip}}

\begin{document}
An equation just to start
\begin{equation}
P+N=S_{d}
\end{equation}
where:
\begin{conditions}
 P     & =       &  notional permeability factor \\
 N     & \sim    &  number of waves \\
 S_{d} & \propto & damage level
\end{conditions}

An equation just to start
\begin{equation}
P+N=S_{d}
\end{equation}
where:
\begin{conditions*}
 P     & =       & notional permeability factor and something 
                   longer that needs to be taken at the next line\\
 N     & \sim    & number of waves \\
 S_{d} & \propto & damage level
\end{conditions*}

\end{document}

In order to have no break after “where:”, here's a variant of the first solution:

\documentclass{article}
\usepackage{array}

\newenvironment{conditions}[1][where:]
  {#1 \begin{tabular}[t]{>{$}l<{$} @{${}={}$} l}}
  {\end{tabular}\\[\belowdisplayskip]}

\begin{document}
An equation just to start
\begin{equation}
P+N=S_{d}
\end{equation}
\begin{conditions}
 P     &  notional permeability factor \\
 N     &  number of waves \\
 S_{d} &  damage level
\end{conditions}
Some text after the equation.

\end{document}

The conditions environment has an optional argument for changing the fixed where:; so, for instance, \begin{conditions}[with:] will use “with:”.

A different version that allows for longer descriptions that needs to be wrapped across lines:

\documentclass{article}
\usepackage{array,tabularx,calc}

\newlength{\conditionwd}
\newenvironment{conditions}[1][where:]
  {%
   #1\tabularx{\textwidth-\widthof{#1}}[t]{
     >{$}l<{$} @{${}={}$} X@{}
   }%
  }
  {\endtabularx\\[\belowdisplayskip]}

\begin{document}
An equation just to start
\begin{equation}
P+N=S_{d}
\end{equation}
\begin{conditions}
 P     &  notional permeability factor with some more text
          so this ends up to break across lines blah blah
          blah blah\\
 N     &  number of waves \\
 S_{d} &  damage level
\end{conditions}
Some text after the equation.

\end{document}

As daleif suggested you could use the align* environment. You get the alignment just like in tables with '&'. Here is an example:

\documentclass{article}
\usepackage[fleqn]{amsmath}
\begin{document}

\begin{align*}
    H_s &= \text{significant wave height, equal to the average of the highest 1/3 of
    the waves}\\
    \Delta &= \text{relative buoyant density, equal to }\rho_r\text{ / }\rho_w - 
    1\text{, where }\rho_w\text{ is the water density}\\
    D_{n50} &= \text{nominal  diameter defined in Equation \eqref{eq:dn50g50}}\\
    P &= \text{notional permeability factor}\\
    S_d &= \text{damage level}\\
    N &= \text{number of waves}\\
    \xi_m &= \text{breaker parameter based on mean wave period }T_m\\
    \alpha &= \text{slope angle}
\end{align*}
\end{document}

This should work for you. This is the output:

If you don't want all equations to be aligned left, you can use the flalign* environment, like suggested here ^[1].

Another option is to use the tabbing environmnent:

\begin{equation}
\frac{H_s}{\Delta D_{n50} } = 1.0~ P^{0.13}~ \left(  \frac{S_d}{N} \right)^{0.2}  \xi_m^P~ \sqrt{\cot \alpha}
\end{equation}
where: 
\begin{tabbing}
\phantom{$D_{n50}\ $}\= \kill
$H_s$\> = significant wave height, equal to the average of the highest 1/3 of the waves\\
$\Delta$\> = relative buoyant density, equal to $\rho_r / \rho_w - 1$, where     $\rho_w$ is the water density\\
$D_{n50}$\> = nominal  diameter defined in Equation \eqref{eq:dn50g50}\\
$P$\> = notional permeability factor\\
$S_d$\> = damage level\\
$N$\> = number of waves\\
$\xi_m$\> = breaker parameter based on mean wave period $T_m$\\
$\alpha$\> = slope angle\\
\end{tabbing}

The first line in the tabbing environment will set the spacing (i.e. tabstop). The \kill command suppresses the line it precedes, otherwise you would have an extra space between 'where:' and the first definition.

Alternatively, instead of \phantom{$D_{n50}\ $} (the longest item on your list), you can use a fixed width: \hspace{3cm}.

This is an old post but all of the solutions require the equation and the explanations to be given separately. I prefer to have a single environment that combines the equation and the explanation of its variables.

The code below defines a new formula environment where you specify the formula as the first argument and give the explanation of the variables as a comma separated list in the body of the environment. For example, the code,

\begin{formula}{P+N=S_{d}}
 P    : notional permeability factor,
 N    : number of waves,
 S_{d}: damage level
\end{formula}

produces

Inspired by egreg's nice answer, there are additional arguments. You can use an exclamation mark if you want a line break after the where: and you can change the where: into anything you like using the (second) optional argument. So

\begin{formula}![where the wonderful notation we are using is:]{P+N=S_{d}}
 P    : notional permeability factor,
 N    : number of waves,
 S_{d}: damage level
\end{formula}

produces:

Additionally, there is a formula* variant that works in exactly the same way except that it does not print an equation number. For example, to display the formula in the OP we could use:

\begin{formula*}!{
   \frac{H_s}{\Delta D_{n50} } = 1.0~ P^{0.13}~ \left(\frac{S_d}{N} \right)^{0.2}
                \xi_m^P~ \sqrt{\cot \alpha}
  }
  H_s: {significant wave height, equal to the average of the highest 1/3 of the waves},
  \Delta:  {relative buoyant density, equal to $\rho_r / \rho_w - 1$, where
             $\rho_w$ is the water density},
  D_{n50}: nominal  diameter defined in Equation (2),
  P: notional permeability factor,
  S_d: damage level,
  N: number of waves,
  \xi_m: breaker parameter based on mean wave period $T_m$,
  \alpha: slope angle,
\end{formula*}

produces:

Notice that it is necessary to enclose the variable explanations that contain commas inside braces.

Here is the full code:

\documentclass{article}
\usepackage{amsmath}
\usepackage{etoolbox}
\usepackage{xparse}

% add a single explanation line - #3 gobbles any extra garbage
\def\ExplainVariable#1:#2:#3!!!{$#1$ &$\;\;=$ #2 \\}

\NewDocumentEnvironment{formula}{ t! O{where:} m b }{%
 \begin{equation}#3\end{equation}
  #2 \IfBooleanT{#1}{\\[1ex]}
  \renewcommand*\do[1]{\ExplainVariable##1::!!!}%
  \tabular[t]{r@{}l}\docsvlist{#4}\endtabular
}{}
\NewDocumentEnvironment{formula*}{ t! O{where:} m b }{%
 \begin{equation*}#3\end{equation*}
  #2 \IfBooleanT{#1}{\\[1ex]}
  \renewcommand*\do[1]{\ExplainVariable##1::!!!}%
  \tabular[t]{r@{}l}\docsvlist{#4}\endtabular
}{}

\begin{document}

  \begin{formula}{P+N=S_{d}}
     P    : notional permeability factor,
     N    : number of waves,
     S_{d}: damage level
  \end{formula}

  \bigskip

  \begin{formula}![where the wonderful notation we are using is:]{P+N=S_{d}}
     P    : notional permeability factor,
     N    : number of waves,
     S_{d}: damage level
  \end{formula}

  \bigskip

  \begin{formula*}{
     \frac{H_s}{\Delta D_{n50} } = 1.0~ P^{0.13}~ \left(\frac{S_d}{N} \right)^{0.2}
                  \xi_m^P~ \sqrt{\cot \alpha}
    }
    H_s: {significant wave height, equal to the average of the highest 1/3 of the waves},
    \Delta:  {relative buoyant density, equal to $\rho_r / \rho_w - 1$, where
               $\rho_w$ is the water density},
    D_{n50}: nominal  diameter defined in Equation (2),
    P: notional permeability factor,
    S_d: damage level,
    N: number of waves,
    \xi_m: breaker parameter based on mean wave period $T_m$,
    \alpha: slope angle,
  \end{formula*}

\end{document}

The basic idea to parse the body of the formula environment using \docsvlist from the etoolbox ^[1] package.

When I've faced this problem, I wrote the package eqexpl ^[1] (since 14 Nov 2018 it is available on CTAN). With it you could add in preamble

\usepackage{eqexpl}
\eqexplSetIntro{where:} % set parenthesis in the left of the first item
\eqexplSetDelim{=} % set delimiter to "="

and then in document

\begin{eqexpl}
  \item{P} notional permeability factor
  \item{N} number of waves
  \item{$S_d$} damage level
\end{eqexpl}

With this in preamble

\usepackage{eqexpl}
\eqexplSetDelim{=}

and this in the document

\begin{eqexpl}[25mm]
  \item{$H_s$} significant wave height, equal to the average of the
    highest 1/3 of the waves
  \item{$\Delta$} relative buoyant density, equal to $\rho_r / \rho_w - 1$,
    where $\rho_w$ is the water density
  \item{$D_{n50}$} nominal diameter defined in Equation (2)
\end{eqexpl}

you'll get

Project on GitHub ^[2] and brief documentation ^[3]

I think you can try fairly easy to use alignat environment:

\documentclass{minimal}
\usepackage{amsmath}
\begin{document}

    \begin{alignat*}{3}
      &A   &= &e^{\pi}  \\
      &AB  &= &\sqrt{x} \\
      &ABC &= & \text{An equation with three letters on the left!}
    \end{alignat*}

\end{document}

Resulting in

In this case, where there are lots of text on the right, tabular with symbol separation would have been sufficient (and simpler):

\begin{tabular}{l @{ $=$ } l}
$P$ & notional permeability factor\\
$N$ & number of waves\\
$S_d$ & damage level \\
$f(x,y,z)$ & distribution
\end{tabular}

Resulting in

It should be possible to replace @{ $=$ } with !{=} since the ! adds space before and after the separator but I can't get it work.

Ps: thanks to siteduzero.com

I created an Excel based solution to maintain symbols definitions, units and equations in an Excel worksheet. This Excel workbook generates LaTeX code to use in your document.

The Excel sheet automatically generates the right code for parameter definitions when you enter an equation.

This sheet can help you with a pleasant workflow in creating the parameter descriptions and saves manual work.

I chose a layout in which symbol units are also added. Here's an example:

I created a Github project to maintain this Excelsheet and accompanying .tex files and make it available to others: https://github.com/johanf85/Symbollist-maintain-Excel