How to put a list and an equation side by side

The enumerate environment would need to go into a vertical box to fix this. There are lots of ways to do this, I think the simplest is to change your column type to p{<width>} and then specify the width; I've used .5\textwidth below to represent 'half of the textwidth'

\documentclass{article}

\begin{document}

\noindent\begin{tabular}{p{.5\textwidth}c}
Known values:
\begin{enumerate}
\item $I_0 = 1nA = 1\cdot 10^{-9} A $
\item $T_1=20$
\item $T_2=0$
\item $T_2=100$
\end{enumerate}
&$I=I_0\cdot (e^{\frac{q\cdot v}{k_{bT}}}-1)$\\
\end{tabular}
\end{document}

Other options include using a vbox, minipage, or parbox, but I think my displayed method is easiest (based on what you have already); explore as you see fit :)

As a side note, to customize any list environment, I would highly recommend the enumitem package, which would allow you to use

\begin{enumerate}[label*=\roman*.]

or, set it globally in the preamble using

\setlist[enumerate]{label*=\roman*}

---

For a vertically centred equation alignment, you could consider using the array package's m column specification:

\documentclass{article}
\usepackage{array}% http://ctan.org/pkg/array
\usepackage{enumerate}% http://ctan.org/pkg/enumerate
\begin{document}
\noindent
\begin{tabular}{@{}m{.5\linewidth}@{}p{.5\linewidth}@{}}
  Known values: \\
  \begin{enumerate}[i.]
    \item $I_0 = 1nA = 1\cdot 10^{-9} A $
    \item $T_1=20$
    \item $T_2=0$
    \item $T_2=100$
  \end{enumerate} &
  \centering $I=I_0\cdot (e^{\frac{q\cdot v}{k_{bT}}}-1)$
\end{tabular}
\end{document}​

What about putting your enumeration and equation into two side-by-side minipages instead of tables? As you prefer alignement of the two elements at their top, I added some top alignment (cf. Understanding minipages - aligning at top).

Unfortunately then it is no longer possible to set the equation as displayed equation because the space above the equation would inhibit the correct alignment. This again could be fixed by putting a \displaystyle within the equation

\documentclass{standalone}
\begin{document}
    \noindent
    \begin{minipage}[t]{.5\linewidth}
        \begin{enumerate}
            \item $I_0 = 1nA = 1\cdot 10^{-9} A $
            \item $T_1=20$
            \item $T_2=0$
            \item $T_2=100$
        \end{enumerate}
    \end{minipage}%
    \begin{minipage}[t]{.5\linewidth}
        $I=I_0\cdot (e^{\frac{q\cdot v}{k_{bT}}}-1)$
    \end{minipage}
\end{document}