Make \cite{my reference} show name and year

The simplest way is to use the bibliography style: apalike, instead of plain. Here is:

\bibliographystyle{apalike} 
\bibliography{References}

---

A more powerful package is natbib. In your case, use citet for textual citations.

\usepackage{natbib}

\citet     #textual citations, print the abbreviated author list
\citet*    #textual citations, print the full author list

\citep     #parenthetical citations, print the abbreviated author list
\citep*    #parenthetical citations, print the full author list

\citealt    #the same as \citet but without any parentheses.
\citealp    #the same as \citep but without any parentheses. 


\citeauthor{ale91}         #Alex et al.
\citeauthor*{ale91}        #Alex, Mathew, and Ravi

\citeyear{ale91}           #1991 
\citeyearpar{ale91}        #(1991)

Here's a solution that uses the natbib citation management package and the plainnat bibliography style. (Note that the plain bibliography style is not well-suited for authoryear-style citations.) As always, run latex, bibtex, and latex twice more to generate the bibliography and compile the document.

\documentclass{article}
\usepackage[round]{natbib}   % omit 'round' option if you prefer square brackets
\bibliographystyle{plainnat}
\begin{document}
\citet{Franklin1999}
\bibliography{my_bibtex}
\end{document}

What you probably want is something like the authoryear citation style of the biblatex package. If you are trying to follow a particular citation style, it would be best to say which in your original question; a biblatex style likely exists for it.

To compile this example, run

pdflatex my_paper
biber my_paper
pdflatex my_paper
pdflatex my_paper

my_paper.tex:

\begin{filecontents*}{my_bibtex.bib}
@article{Franklin1999,
        author = {Allen Franklin and Karjalainen Risto},
         title = {Using genetic algorithms to find technical trading rules},
          year = 1999,
        volume = 51,
         pages = {245--271},
  journaltitle = {Journal of Financial Economics}
}
\end{filecontents*}
\documentclass{article}
\usepackage[style=authoryear]{biblatex}

\addbibresource{my_bibtex.bib}

\begin{document}
Important result found by~\cite{Franklin1999}.
\printbibliography
\end{document}