What you are referring to is either a cable or DSL connection. In both cases you are connected to the internet with a modem. In fact you still need a modem even with a dial up connection but dial up is considered to be the slowest.
DSL and cable are very close in speed but DSL connection is superior to cable in the sense that you don\'t share your broadband with other users. This way internet traffic won\'t affect your speed one way or another. Pricewise DSL is more money but not available in all the States yet.
Your sentence:
My internet has a cable connection. (= you have to have a modem to run a cable connection)