jmc

Let's start by finding the units digit of small powers of 2. $$\begin{array}{rl}{2}^1& =2\\ 2^2& =4\\ 2^3& =8\\ 2^4& =16\\ 2^5& =32\\ 2^6& =64\\ 2^7& =128\\ 2^8& =256\end{array}$$It looks like the units digit repeats every time the exponent is increased by 4. The remainder when 2010 is divided by 4 is 2, so $2^{2010}$ has the same units digit as $2^2$, which is $\boxed{4}$.