Typical requirements for dial-up networking:
Assuming you've got the first two items taken care of...
You'll need to contact your local telco company for this; in the US this will usually be either one of the "baby Bells" like Qwest, or Verizon. Check your local telephone directory if you're unsure. Typical cost for a basic land line is around $25/month with all the taxes. About the only extra you might want is call waiting if your modem is V.92 (see below) and you don't want to miss incoming calls. Don't be surprised if the telco asks for a deposit and takes a week to activate your line...
These can be picked up almost anywhere; chances are you already have one. That said, the typical flat phone cord isn't ideal as there is barely enough copper to carry a signal and the pair isn't twisted. Best option is to get some connector crimps and modern communication cable and make your own cord. And don't stop at the phone jack; if the wires running to the customer service box are really old replace them too.
"Modem" stands for MOdulator-DEModulator and is a device that modulates/demodulates an analog carrier signal to encode/decode digital information. Modems allow digital computers to communicate over analog telephone lines. In the case of dial-up, the modem is of "narrow band", ie. it operates within the traditional POTS frequency band of 0 to 4 KHz.
There's basically two types:
Modems can either be HARDware or SOFTware driven. Software modems (softmodems) require OS-specific drivers to function whereas hardware modems (hardmodems) rely on self-contained controller chips for functionality and are therefor generally OS-independent.
Users in need of a hardmodem should look for terms like "controller-based" or "supports DOS/Linux/Mac OSX"
Modern telco networks switched to digital equipment in the late 1990s; the V.90/V.92 modem protocols were designed to take advantage of those telco upgrades and make use of hardware compression to increase transmission rates up to 320.0 kbits/s.
Most SDF Dialup access numbers support at least V.90. Older modems may work as well but at slower speeds - ie. V.34 modems operate at 28.8 kbits/s
Internal modems are easy: just install it in an open card bay, plug one end of the phone cord into the the modem jack, the other into the phone jack in the wall/baseboard. External modems work similarly except you'll also need to connect the modem to the computer via either a serial or USB cable, and of course plug in the modem's power supply. Once all the cables are connected, power up and configure the dial-up PPP network connection (see the Windows and Ubuntu examples).