Depends on the context.

It could mean amusing them - like how you have to play with a child to keep them entertained rather than just talk to them or hang out.

It could also mean having someone in your home, though that is more archaic usage. Entertaining used to mean hosting dinner parties and the like.

In the more archaic usage I've heard it used for listening to the thoughts of someone you normally wouldn't because they are beneath your station or the enemy or......

Its providing enjoyment or something to do. If I say I'm entraining guests at my house it's typically I'm hosting something that may include dinner, drinks, games etc.

Ah yes, to "entertain an idea".

Good add.

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"Entertaining" usually means some event that you put effort into organizing. You invite the guests ahead of time, make some food, and often have a focal activity (watching a movie, doing a craft, gift exchange, etc.)