Here's an excellent video by AronRa exploring the classification of dinosaurs (and their non-dinosaurian relatives).

So what about the dinosaurs?

Birds are dinosaurs.

Pterosaurs are not dinosaurs, but they share a close common ancestor.

Ichthyosaurs are not dinosaurs.

Mosasaurs are not dinosaurs, instead they're related to today's monitor lizards.

Plesiosaurs are not dinosaurs either.

Dinosaurs are not lizards.

Many dinosaurs have been discovered to have feathers (yes, Velociraptor is fluffy!)