A BBS was a Bulletin Board System. Basically a computer with a modem and software that would receive calls instead of dialing out. You would call a BBS computer with your modem, and you could then upload or download files, read and post messages, play games, stuff like that. That computer didn't have any connection to the rest of the world other than the modem line, so whatever you posted or uploaded was isolated to that one computer. People got clever and set up their BBSes in networks so that every morning, say at 4:00am, one BBS would dial another and synchronize their messages and files, so over the course of several days the data would slowly propagate across the city until all the participating BBSes had a copy of it.
Anyone with a computer, a modem, and a spare phone line could set one up. It was then a popularity contest to attract users to your system instead of someone else's. Certain BBSes were known for having the best repository of files (be they "demos", porn, warez, or legitimate collections of shareware), some for their games, some for programming, etc.
Think of dial-up like calling a corporation's call center, where you get connected to the first available representative (out of possibly hundreds) and each representative is logged into the same computer system so they can each give you the same information as any other. While calling a BBS was more like phoning up your friend. You could share the latest gossip and then he'd call up his friend and so on. It was far more intimate than what we have now, and not a lot unlike social networking (just much slower!)
Certain BBSes were so fancy that they had multiple phone lines (I was signed onto one that had 12 lines going) and you could actually interact with other users who were dialed in at the same time as you. You could set a message in your "WHO banner" to show everyone (like a Facebook status update) and you could send instant messages to other users (like ICQ and MSN messenger which were still years away from being popular). Everything old is new again!
Remember this was all text-based, so no graphics (but the fancier ones had color!) and file transfer times were very slow -- to download a 1 megabyte file could take 5-10 minutes. But it was all such good fun. Ah, memories.
And MUDs? That stands for Multi User Dungeon and it is basically a text-based RPG. Think World of Warcraft minus the graphics. The system would describe things to you ("You are in a room. There is a table here. A sword lies on the floor. Exits are east and north.") and you would type instructions ("get sword", "north", etc.) You could battle with computer-controlled characters or other users. Crude, by today's standards, but surprisingly immersive and addictive!
