Hard phones are physical pieces of hardware that plug into your network typically through an ethernet connection. They are the best type to use as they have no dependency on a computer. This means that they will work whether your computer is turned off or does not exist at all. Also, it is very nice to use a regular handset style phone instead of a microphone and computer speakers. One potential disadvantage is that your home network needs to have an unused ethernet port in which to plug the phone in.
Cisco ATA 186/188
Linksys PAP2T
Linksys SPA-2102
Linksys SPA-942

Soft phones are simply a piece of software that is loaded onto your computer and uses the computer's soundcard as a user interface. Most people start with a soft phone because of their low price (typically free) and availability (easily downloadable). The down side is that you do need a computer and it is often more difficult to use a microphone/headset than a traditional handset.

Xten X-Lite / X-Pro
SJ Labs - SJPhone

Firm phones are a combination of a soft and hard phone. They are handsets that plug into your computer which utilize intelligent drivers running on the computer to communicate with the network. They are a good compromise, both inexpensive and the ease of use with a handset. Most of these phones use a simple USB connection to interface to the computer. The down side is that they do need software drivers and a computer to work.

USB Phone
Kerry USB Phone