When you're entering a dungenon and it shows "Random dungeon name First Level", does that mean there is a second and/or third level as well? I'm pretty sure I go through the entire dungeon but don't see a second level on a lot of them.
All dungeons that have "Level x" in the name will have at least 2 levels.
Random dungeons that have "Level x" will always have only 2 levels (and normally a resplendent chest on the second level), while story dungeons (like the crater in act 3, or the Cathedral in act 1) can have more (I think 4 might be the highest).
As Mr.Smooth already said, there are always at least 2 levels on a dungeon with "Level x" in the Name. But not every Dungeon have this in the name, for example the Caverns of the Graveyard on Act 1.
You can ensure that you reached the last level of a random dungeon when there is a stone to port you back to the entrance of the dungeon (I don't know at the moment what they are called.)
The entry to the second level might be in some hidden path, be sure there isn't a little dark spot on the map when you don't find it the first time.