I like drama movies. More than pure comedies most any day. But a mix of comeday and drama can also be fun. For dramas, I like things like:

The Wrestler
The Godfather
Letters from Iwo Jima
Unforgiven
Million Dollar Baby
Mystic River... alright, most any movie Clint Eastwood has directed lately

I also like dramatic documentaries, like:

Anvil: The Story of Anvil

I also like challenging dramas, like:

Dogville (never want to see it again)

Really, if you can relate to characters, or the plot can make you think... then the movie has succeeded.