A Realm isn't quite the same as a universe. One universe has 9 Realms and many more dimensions.
Realms
In Marvel terms, the 616 universe (which is where the main stories were set, before the current Secret Wars storyline) has 9 Realms:
- Alfheim - home of the light elves
- Nidavellir - home of the dwarves
- Jotunheim - home of the giants
- Svatalfheim - home of the dark elves
- Hel - the realm of the dead
- Muspelheim - home of the demons
- Vanaheim - home of the Vanir
- Midgard - where the humans live.
- Asgard - home of Thor etc..
Our realm is Midgard. This is pretty much the Norse mythology view of things.
Illustrated here in an old Thor comic:
However, during Original Sin, a tenth was added:
- Heven - home of the Angels.
Here's a map from Thor v4 #7, with Ygdrassil in:
Other Dimensions
On top of that you have other dimensions such as Limbo (e.g. Magik of the X-Men) and other demon specific realms (from Doctor Strange stories). See
wikipedia for a list. Highlights include:
- Limbo - Home of Belasco, featured in the X-Men books and linked to Magik
- The Mojoverse - home of the X-Men villain Mojo
- The Microverse - where Ant Man shrinks to
- The Negative Zone - featured prominently in Fantastic Four books
(The "realms" of the Thor books are basically interconnected dimensions)
As to how many universes in the multiverse - see How many Marvel Earths (Universes) are there?.