Previewing Animation in Maya

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In Maya, animation previews are called "PLAYBLASTS"
Go to:
Window > Playblast >

Or:

Right click on the timeline and choose "Playblast > " from the popup menu.
Choose your options for the size, compression, location, filename, etc.
Encoding is a bit tricky at the moment. No good codecs appear in my dropdown... 

Recommended playblast settings: 

IYUV, 50% quality, 640 x 360 100% size

For short cycles, this usually results in a nice compact file  <10 MG.

Autodesk Help has a more detailed explanation here:
Playblast animation at Autodesk Help