What tool do you use to get the interlocking puzzle piece kind of thing on the drawers?

The "interlocking puzzle pieces" are called dovetails because the look just like the tail of a dove. I use these for all my drawers for two reasons. One, they look good, and two, they are arguably the strongest joint there is when dealing with a linear pull. There are a few types of dovetails, this one in particular is called a half-blind dovetail. Reason being you can see it from the side, but not the front. There are also through dovetails which can be seen from both sides, and sliding dovetails. I'll put up examples of each in the future.

As for your question, I use a dovetail saw (similar to this http://www.leevalley.com/US/Wood/page.aspx?p=48032&cat=1,42884), chisels, wooden mallet, and, on bigger dovetail pieces, a modified piece of bandsaw blade (teeth cut off and back sharpened like a knife). An easier way to do these is just to use a router with a dovetail jig and dovetail bits. It's a neat setup, but the angle of the dovetail is limited due to the physical limits of the router bits. I'll elaborate more on how I do it by hand if you'd like.

It's so nice I thought you had to do it by machine! I know very little about woodworking.

Anything that can be done by machine can done by hand, and there are some things done by hand that cannot be done by machine. I'll put up a chair I did as an example. Keep asking questions, I'll learn you.