You will need to have at least version 2.1 of Android, which you can find by having a quick glance at Settings > About. Skype has tested the software on HTC and Motorola devices but should work on a wide variety of platforms. They have reported some issues with the Samsung Galaxy S (which they are looking into).
Once Skype is launched it works much like any other Skype for mobile clients. Your contact list and call history is synchronized with your Skype login giving you the same contacts as on your Desktop. Calls may be made over Wifi and 3G, except in the United States where calls are wifi only.
Video calling does not work on the Skype for Android app.
Skype calling is free to other Skype users. If you have Skype credit you may dial out to any number at regular Skype rates, and if you have a SkypeIn number calls will ring to your device. This is a great way to offset your mobile minutes which can be an order of magnitude more expensive than SkypeOut minutes, especially for international calling.
Skype has a video which shows how their mobile app works.
Try it out today and let us know how you find it!