I am in the process of making some drawers for one of my cabinets. I have a rough idea of how I will do it, but if someone who has made this kind of drawer could tell me what all I will need it would be very helpful for me.
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Adam Jaskiewicz covered the basics, but I'll add a few comments. I've built exactly one "furniture grade" drawer, but several "shop grade" drawers. And I've never done a dovetail joint. So take what I say with at least one grain of salt.
The basic drawer box is four sides with a groove cut into the inside for the bottom. Depending on use, you can use 3/4 plywood or 1" dimensional lumber. You could probably get away with 1/2" plywood for small drawers (for example, in a dresser), but I think the 3/4 is a better choice if you have anything heavy inside. The can be 1/4 plywood, and it's not glued to the rest of the drawer.
You can use either wooden slides (basically, a groove cut into the back of the drawer and matching strips of wood that run front to back on the piece), or a side-mounted metal slide. I recommend the metal slide, particularly for a kitchen cabinet, which will be fully extended. I really can't recommend using wooden slides: they may be more "pure," but they're a lot less functional (and yes, for my one furniture-grade drawer, I used wooden slides; it doesn't get a lot of use).
Here's a page that describes the basic cuts for a drawer: http://www.knottyplans.com/index.php?page=200422&ref=kp32ap98f (it was one of the top hits on Google that looked reasonable).
The biggest issue that you'll face when building a drawer is how to join the front and sides. This joint will take all of the stress of opening the drawer.
The traditional approach is to use a dovetail joint. If you have a router and a dovetail jig, this is a great approach. It's particularly good if you're making a lot of drawers, and can set the jig once for all of them. You can also make dovetails with a backsaw (or better, a nokogiri saw), but you'll need lots of patience and practice.
Although a dovetail joint is traditional, a finger joint is almost as strong, and you may can make it using a table saw (assuming that you have a dado blade and a vertical jig). There are several other types of joints of varying strength; Fine Woodworking did an article on them several years ago.
If you go the dovetail route, the front panel of the drawer should be the width of the opening minus whatever you need for the slides. In furniture applications that have slides, there's typically a "face" front that's attached to the structural front, and covers the gaps. The alternative is a half-blind dovetail.
However, if you're looking to create drawers that live inside a cabinet (and are therefore hidden behind a door), you can get away with an alternate joint, in which the front panel is held between the side panels, and is screwed in place:
||==front==|| || ||
In this approach, all of the strength in the joint comes from the mechanical fasteners. So don't skimp: go thick (#8 or better), long (1-1/2 or better), and repeated (one screw per inch). You could also use finishing nails, although they do not provide the flex resistance of screws. Do not rely on glue; there is no surface area for the glue to bond (as far as I'm concerned, end-grain is useless). Oh, and pre-drill the holes.
Well, it's like any other drawer, but the front is sized to fit completely into the cabinet rather than against the front of the frame.
It looks like that particular drawer has slides that mount to the underside of the drawer, like these (not an endorsement, just an example with a good picture).