What was that plastic insert called for record albums?

Answer 1

If you are talking about 45 rpm records, 'singles', the plastic insert was called an adapter, or a spindle adapter. They allowed you to play 45's without the awkward 45 spindle you would otherwise have to insert.

Answer 2

The plastic insert for record albums is called a inner sleeve or a poly-lined sleeve. It is used to protect the record from dust and scratches while stored inside the album cover.