Dial up is very slow. Basically, a dial up connection uses the analog signal of your phone line to transmit and receive data. If you pick up a phone that's on the same line as your computer while it's online, you'll hear the modem talking to the Internet service provider. Since the phone network was not designed for this type of connection, it's very slow.
DSL is a form of broadband that also uses your phone line, but the big difference is that this is a digital signal and is much faster. DSL doesn't use the phone line the same way a dial up modem does, so you can talk on the phone at the same time your computer is online. DSL is vastly faster than dial up.
Cable is another digital technology, but it uses your TV cable to send the Internet signal. Cable is usually, but not always, faster than DSL.
Another option in some areas is wireless. Companies like AT&T and Verizon sell "aircards" that connect your computer to the cell phone network and allow it to get Internet that way. Aircards are usually portable and will give you Internet access whereever you have cellular coverage. These are very popular for use with laptops as they allow you to have an Internet connection essentially anywhere. These are still much faster than dial up but usually not quite as fast as Cable or DSL. One drawback to aircards is that they typically give you a certain amount of data that you are allowed to transfer and if you go over this amount, they will charge you overage fees which can be pretty expensive.
There is also another type of wireless which is often called WiMax. WiMax modems are similar to aircards, but they are typically larger and designed to stay in one place. WiMax is usually about as fast as DSL. One drawback to WiMax is that you can't get it everywhere and even if you can get it, you may experience signal problems since it is a wireless technology. Just like on your cell phone when you have no bars, with WiMax if you have a bad signal you have no Internet.
Another type of Internet is delivered by fber optic cable. This is usually the fastest type of Internet, but is only available in a few areas.
Cable and DSL are by far the two most popular types of Internet these days, so I would go with one of those if possible.