How can I reduce my code when I used \addplot [black, mark = *] coordinates many times?

As Rmano pointed out, one \addplot command is enough, i.e.

\addplot [black, mark = *,only marks] coordinates {( 0, 2)(-1, 1)(1, 1)(-2, 10)(2, 10)} ;

Another option would be use the samples at key for \addplot, and let pgfplots calculate the y-values:

\addplot [black, mark=*,only marks,samples at={0,-1,1,-2,2}] {(x^4 - 2 * x^2 + 2};

If you want to shorten the code more, you can first declare a function with e.g.

declare function={Y(\x)=\x^4 - 2 * \x^2 + 2;}

and use Y(x) in the different \addplots. You can also make the code drawing the dashed lines shorter using \pgfplotsinvokeforeach.

Complete code, the output is the same as in your image:

\documentclass{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.9}
\usepackage{fouriernc}
\begin{document}
\begin{tikzpicture}
\begin{axis}
    [
    declare function={Y(\x)=\x^4 - 2 * \x^2 + 2;},
    axis lines = center,
    xlabel=$x$,ylabel=$y$,
        domain=-3.5:3.5,
        ymin=-0.85,
        ymax=11,
        xmin=-3,
        xmax=3.5,
    samples=100,xtick distance=1,
ytick distance=2,unit vector ratio*=1 1 1,
    width=11cm,
    grid=major,
    grid style={gray!30}
    ]
       \addplot [black, thick] {Y(x)};
       \addplot [black, mark=*,only marks,samples at={0,-1,1,-2,2}] {Y(x)};
       \node at (axis cs:-0.25, -0.25) {$O$} ;

       \pgfplotsinvokeforeach{1,2}{
       \draw [blue,dashed] (axis cs:-#1,0) |- (axis cs:#1,{Y(#1)}) -- (axis cs:#1,0);
       }

       \end{axis}
\end{tikzpicture}
\end{document} 

Hmmm... quick'n'dirty --- define a command:

\newcommand\splat[1]{ \addplot [black, mark = *] coordinates{#1} }

And then

        \splat{( 0, 2)} ;
        \splat{(-1, 1)} ;
        \splat{(1, 1)} ;
        \splat {(-2, 10)} ;
        \splat {(2, 10)} ;

(millions of variations possible...)

Or in this case you can just use the only marks keyword:

\addplot [black, mark = *, only marks ] coordinates { ( 0, 2)
        (-1, 1)
        (1, 1)
        (-2, 10)
        (2, 10)};