I know spying on somebody means to watch somebody secretly, but is there a term that means to listen to somebody secretly? For example, you are in a room talking with someone and there is someone else outside the door listening to your conversation without you knowing.
A usage note at the first link points out:
To eavesdrop usually implies the listener is purposefully trying to hear the conversation of others. To overhear is more often accidental.
In addition to the very fitting answers provided by jwpat7, there is also the phrasal verb listen in.
listen in (phrasal verb) listen to a private conversation, often secretly
I usually hear it used with the word on:
Maybe I should have minded my own business, but I couldn't help but listen in on Rob and Amanda's conversation.
Also, if the eavesdropper is using electronic equipment to monitor the conversation, an oft-used word is bug, which can be used as a verb (to conceal the device) or a noun (for the device itself):
The spy was not aware that someone had bugged the room.
The spy planted a bug in the room.