How does HOD differ from L? If $L$ is the class of constructible sets and $HOD$ is the class of hereditarily ordinal definable sets, in what ways do they differ?
I know that they do as $L$ is absolute in transitive models of $ZFC$ and $HOD$ isn't.  But does $HOD$ always contain $L$? Is there a simple example of a set that can be in $HOD$ but not in $L$?
 A: $\mathrm{HOD}$ always contains $L$ because any inner model contains $L$, by absoluteness. 
How easy it is to exhibit a difference really depends on your background. For instance, $0^\sharp$, if it exists, is a real that always belongs to $\mathrm{HOD}$ but is not in $L$. 
If you are not too comfortable with large cardinals, but know forcing, you may enjoy proving that any set is consistently in $\mathrm{HOD}$. This gives you many examples. The argument is actually simple: Just code the set using the continuum function. 
To further simplify matters, imagine the set, $X$ is a set of ordinals, say $X\subset\alpha$. By a preparatory forcing (collapsing a few cardinals if necessary), you may assume that a long initial segment of the universe satisfies $\mathsf{GCH}$. Now force adding Cohen subsets of different cardinals so that in the extension $2^{\aleph_{\beta+1}}$ is either $\aleph_{\beta+2}$ or $\aleph_{\beta+3}$ for all $\beta<\alpha$, with $2^{\aleph_{\beta+1}}>\aleph_{\beta+2}$ if and only if $\beta\in X$. This allows you to define $X$ in the extension as the set of $\beta<\alpha$ such that the continuum hypothesis is violated at $\aleph_{\beta+1}$. This shows that $X$ is ordinal definable and therefore, since it is a set of ordinals, it is in $\mathrm{HOD}$. [I use only successor cardinals to do the coding so you can arrange this situation by Easton forcing.]
(For general sets $X$, not necessarily sets of ordinals, code the transitive closure $T$ of $\{X\}$ as above, and then check that $X$ is definable from $T$.)
The argument can be pushed further, using class forcing. The result is that, no matter what it is to begin with, the whole universe $V$ can be made the $\mathrm{HOD}$ of an extension. With some (serious) additional work, one can use this idea to show that $V=\mathrm{HOD}$ is consistent with (extremely) large cardinals. 
[For moderately large cardinals, one can prove stronger results, by exhibiting canonical inner models of the large cardinal assumptions under consideration. These canonical models "generalize" $L$ and always satisfy $V=\mathrm{HOD}$ and in fact are subclasses of (true) $\mathrm{HOD}$.]
