Just because no-ones said yet how to start this off for the uninitiated...
To obtain birch bark the best time to do it is in the spring, when the tree is beginning to show life again, and won't mind losing a bit of the extra protection it put on for winter.
First, find a birch tree with a large smooth section of trunk with few nots, burrs, hole etc (harder than it sounds!). With a sharp knife, cut into the tree (not very deep) in a straight vertical line. Use the tip of the knife to lift this away a bit to ensure you've gone deep enough, if not cut a bit deeper. Its safets to cut this way as a vertical cut doesn't cause any damage to the tree if you go too deep (whereas a big horizontal cut could cause minor ringing damage).
Then, move round the tree, and cut an identical vertical cut. Finally, cut horizontally between the two vertical cuts - you should now have a square of birch bark that you will use for basketry/weaving.
Repeat this on different trees to obtain enough bark - remembering not to cut too deep, or take too much bark from any tree, particularly diseased or damaged ones.
As has already been said, take these squares of bark, and place them back to back (I generally use clothes pegs to hold them flat) and if you like put a weight on top to keep them flat. For making containers, there are two main ways.
One is to roll the bark up into a cylinder to make a tube. Then take circles of bark larger than the diameter of your cylinder, and draw a smaller circle just fractionally bigger than your cylinder on this circle. Cut 'tabs' in towards this inner circle, and fold up these outer tabs. Take strips of birch bark and weave these in and out of the tabs to form 'sides' to the lid, securing them where they overlap with a couple of stitches. Alternatively, fold the extra length of the tabs back over the woven-in strips and weave the tabs into these strips to secure. Without a picture this might be hard to explain

The height of your lid depends on the length of your tabs and the number of strips used. Make two of these lids, and use them on both ends of a tube, stitching one lid to the tube if necessary to make a base.
(Note, throughout this, you could use glue instead of stitching, I just prefer stitching with home-made cordage for a traditional effect).
The second method is the one pictured at the top of this thread - this is a 'circular' basket, but you can more easily make a square one.
To make either, begin by weaving a flat mat of strips of birch, and ensure you leave plenty of unwoven excess hanging off in all directions. The woven flat part is your base, and the shortest of the excess will determine the basket height.
A circular basket is difficult - you have to begin weaving together the strips from adjoining sides, at 45 degrees to each other, and work round til you complete the basket. This kind is best if someone shows you how
A square basket is much easier - simply fold all the excess up at each side so you have 4 vertical sides - then take new strips of bark, and weave these horizontally round in and out of the side strips, alternating the weave for each strip you add. When you get high enough, fold the vertical strips over the top of the top horizontal strip, and weave these back in to the horizontal strips, opposite to the way you weaved on the way up. This foldover is also often done for circular baskets (and for the lids in the cylinder method described above) and can be seen in the image at the top, where the lip of the basket is rounded.
Hope this all makes sense! :?:
EDIT: I forgot to say, have a large number of small washing line pegs to hand - these are infinitely useful for holding woven sections/strips in place so you can weave one bit, then peg it, and move on to another without your original work becoming undone... And if you want to practice, try using thin cardboard or thick paper first, save wasting your precious bark!
Second EDIT: I forgot to say, whenever you are weaving, you can add in new strips of bark by overlaying two strips and weaving with both together for a section til they grip together - this is useful when you can't get many long strips, but want to make a high-sided basket.