In this answer talks about rip and crosscuts and that one saw is better for ripping than one that is more suited for crosscutting.
What is the difference between the 2 cuts and why would a saw-blade be more suited for one or the other?
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With the rip cut, you cut along the grain; while with the cross cut, you cut across the grain.
Cutting along the grain is a very easy cut; even before you had mechanical saw, you had saws with few but large teeth so you cut as fast and as straight as possible. Essentially, you "rip" the wood apart, like you can split it with an axe, except you'll get a straighter cut as you may still cut some of the wood fibers. This cut has a tendency to bind the blade as the wood fiber relax.
Cutting across the grain is much harder for the saw (you need to cut a lot of fibers), and you typically use a saw with smaller teeth, but many more of them.
One other reason why you have more or fewer teeth has to do with the amount of saw dust you need to evacuate after each cut. As the rip is easier, you'll need more space between the teeth to carry it out.
Image comes from http://www.rockler.com/how-to/blades-101/
Pascal Belloncle has already given a good explanation of why there are two different kinds of blades, but what is the practical aspect of it?
All cuts are done on the same piece of wood with a ryoba saw (one side rip, one side cross).
You can immediately see that the rip cut with the rip blade goes twice as deep (both cuts were done pulling the blade along its full length 3 times). Also, you can immediately see that the cross cut with the cross blade comes out nicely whereas the cross cut with the rip blade is... well, far from perfect.
Thinking back to something Roy Underhill said on The Woodwright Shop, probably 30 years ago, I thought that the teeth of a crosscut blade were angled perpendicular to the stroke of the saw, so as to work like a knife blade slicing through the wood fibers. The rip blade teeth pointed straight down, so as to work like a wood chisel splitting the wood fibers apart. Granted he was talking about hand saws, not power saws. Amazing that I still remember that after all this time.
To precisely answer the question: A rip cut is a cut along the longest dimension of the board or sheet. A crosscut is a cut along the shorter dimension of the board or sheet.
An example of a rip cut would be cutting a 4 foot by 8 foot sheet of plywood into a 3 foot by 8 foot sheet. In that case you would be ripping 1 foot off the sheet.
An example of a crosscut would be cutting a 1 inch by 6 inch by 6 foot long board down to 4 feet in length.
The differences in blades used for these types of cuts are described by Pascal's answer.