Pixar Animation Studios was a company who wanted to make films, they needed money to finance and distribute their films, so they teamed up with Walt Disney Pictures to do that.
My understanding is that up until Toy Story 2, they were still two separate companies teaming together. But then they had a big hoo-hah and almost broke up, because Disney Pictures wanted to pretty much eat them and make Pixar disappear. But then — after lots of arguing — Disney bought Pixar, but promised to give Pixar creative control and their name still on the films they made. So films made by the company Pixar are not just Disney films, they are Disney Pixar films — films made by Pixar Animation Studios, funded and distributed by The Walt Disney Company.
2. Disney Classics
Disney Classics are the — mostly annual — feature length animated releases made by Walt Disney Animation Studios. It started with Snow White, which Walt Disney re-mortgaged his house to make, it was the most expensive animated film ever made for its time. Since Snow White, Disney Animation Studios have — most years — released an animated feature length film, and that is what a Disney Classic is. The most recent is Frozen.
Walt Disney Studios/Pictures make other films too like Mary Poppins and Enchanted, they are Disney films, but not Disney Classics, because they are not the films made exclusively by Disney Animation Studios, they are just made by Walt Disney Studios/Pictures [not Walt Disney Animation Studios].
By now The Walt Disney Company owns loads, they’ve got the Disney Channels, Buena Vista which is their distributing company. They’ve bought Marvel, Star Wars, The Muppets — loadsa stuff.
That’s my understanding of it, but it does get a bit confusing because Disney Classics was a term also used to market the videos, some of which were actual Walt Disney Animation Studios films, but some were things like Pete’s Dragon.
Here‘s a list of those Disney Classics which should officially be called Feature Films of Walt Disney Animation Studios.
[Laura and Malcolm asked me this question on Facebook today. This answer was — embarrassingly — mostly off the top of my head, so there might be some holes in it. Correct me if I’m wrong.]