A dialup modem, a phone line, a service provider, and an account. Your computer may already have a dialup modem in it; look on the back for phone jacks or check your System Information (or equivalent). If your service provider is any good they'll be prepared to give you instructions for setting up and configuring your dialup connection.

For my first three months online I tried not to tie up my only phone line during the hours when I was most likely to be getting calls. Finally I gave up and got a second line just for the computer. I pay more per month for the second line so that I'll have unlimited local calling time; without that, my phone bill would skyrocket.

It helps a lot if you have access to a working internet connection, even if it has to be at the library or something, so you can look up answers to any questions you might run into.
On your computer, you just set up the DSL connection and use it in place of the dialup one. If your provider didn't offer both DSL and dialup access, you might have to change providers.
It's a lot better than no connection. I've been online for 10 years now, using nothing but dialup on my own computers. Occasionally I'll get on someone else's computer and notice how much faster their DSL or cable connection is than what I have, but I'm used to the pace of dialup and it doesn't bother me. If I were big on videos or music, that would be different.

I've also overheard cable customers' side of phone calls to their cable providers as they get talked through restoring their cable connection. That's when I feel best about having dialup; once it's working, it tends to stay working indefinitely except for outright equipment failures.

I'm lucky to live within half a mile of a major phone company facility so I regularly get connected at around 50 kbps. I used to visit someone who lived 7 or 8 miles from town and was lucky to get 30 kbps.
I bought this computer used with an inexpensive no-brand modem already in it. I thought I'd replace the modem with a new, name-brand one that I could trust. For almost four years, whenever the phone line got a bit noisy I kept losing the "receive" side of the connection and having to redial. I suspected everything but the modem. Finally, setting my system back up after a hard-drive failure I had so much trouble reinstalling the name-brand modem that I went back to the cheapo. I've had good connections ever since!