As a noun:
A "plug" is something you insert in something else. Most often to fill a hole. "He put a plug in the hole so the water won't leak out." Or it's the end of an electrical cord that goes into the wall socket.
As a verb:
"Plug" means to use a plug to fill a hole.
"To plug in" can mean one of two things. Literally, it means to insert the plug of an electrical device into the wall socket or another operation similar to that with pieces that fit together in a designed way. It can also mean to bring someone into a discussion, especially by telephone, as in "let's plug Nicolas in on our conversation."
"To plug away" means to work steadily at a long project, as in, "I plugged away on the budget all afternoon."
To "plug up" means the same as just "plug"