How do I use the field API? Someone told me that most of the time I will probably want to use the field API (CCK module in D6), rather than adding custom tables or modifying nodes by myself. How do I use the field API? Could anyone can make me an example?
If you've ever used CCK in D6, the Field UI will be rather intuitive. And unless you need to do very custom work, you won't need the Field API at all, just the Field UI.

But if you do need custom widgets or formatters, take a look at the Examples module for some very comprehensive usage of the Field API.