Necessity is the mother of invention. Modems are am ingenious hack that enable digital communication over an inherently analog medium, the existing phone network.
Modems are limited by Shannon's Law, which, on POTS lines in the USA, limits throughput to (8,000 samples per second) * (8 bits per sample) = 64 kilobits per second.
They are further limited (in the USA, at least) by the Federal Communications Commission (FCC) in that they cannot achieve connect rates of more than 53,000 bps due to FCC-imposed regulations regarding the maximum allowable voltage on telephone lines. Due to less-than-ideal line conditions, however, 56K modems generally connect from 42-52 kilobits per second (kbps).
There are three main types: the "classic" modem, the HCF modem, and the HSP modem.
The classic modem has three main components: a controller, a phone line interface, and a digital signal processor (DSP). They generally cost US $50 or more. I know US Robotics makes these.
A HCF modem, which is less expensive than a normal modem, gives the host CPU the job of controlling the modem's function, while the DSP is still handled by the modem's chipset. Often referred to as 'Winmodems' since the controller/driver software is generally only available for Windows or Mac. Probably the most common form of modem today, these are available in a variety of interfaces (PCI and USB) and can be used on different platforms, mostly PC or Macintosh These usually cost from $20-$50.
An HSP modem (also called softmodem), the cheapest of all, is the logical extension of the HCF modem. In this modem, the tasks of both the controller and the DSP are given to the host CPU, which are relatively insignifcant ones given the computing power of modern CPUs. All that remains on an HSP modem is the physical phone line interface. These are often (though incorrectly) referred to as Winmodems as well, since they have similar shortcomings. These cost $5-$20, and they usually obtain the same connect rates as the other two modem types, which is great, as long as you're using Windows (or if you're lucky enough to be able to make Linux/BSD work with it). These are generally sold as generic brands, but always(?) contain a PCTel chipset and are always PCI.
Also, from what I've seen, HSP modems never have transformers. If your modem has one, you can be pretty sure it's either a normal modem or an HCF modem.