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}