'Dial-up' is a way accessing the internet on a computer via the public telephone network, using a device known as a modem. Technically a dial-up connection works in much the same way as a normal phone call, except that it connects computers at either end instead of two people. It's only accessed when that 'phone call' is made, and each time you access it, you're charged, which makes it quite expensive. In addition, because dial-up uses normal phones lines, its speed and efficiency can be compromised.
'Dial-up' has been almost completely abandoned in favour of broadband.