The word 'modem' itself is actually a acronym for the two words 'MODulator' and 'DEModulator'. Simply put, a modem is a translator. That is, it takes the digial signals coming from the computer and changes them into analog signals so that they could travel along telephone wires. Vice versa, it takes analog signals coming from another computer at the other end of the wire and translates that into digital signals so that your computer could understand it. Both have their advantages and disadvantages.
External modems are easier to install and replace.
Because they have signal lights on them, one can see if information is still being transmitted or received. They also can be swithed off independently of the computer. They are a little more expensive than internal modems and require the purchase of an additional cable modem. Because they are external to the PC, they need an extra plug and power source in order to function. On the other hand, because they are internal, it is a somewhat more complicated process to install and configure them.