If you're knitting from someone else's pattern, it's likely they had a good reason for choosing the ribbing they did and the method used to knit it. A particular rib may flow nicely into the cable in the rest of the project, or a fancy two-color rib coordinates perfectly with the rest of a stranded knitting project, for example.
But if you're designing something for yourself, does it matter what kind of ribbing you use? Personal preference plays a big role in what ribbing you choose, but there are some other considerations.
What is Ribbing?
Ribbing is the general term for any knitting pattern that results in vertical columns of knit and purl stitches. Single Rib or knit 1, purl 1 ribbing is one of the most common types, along with Double rib or knit 2, purl 2 ribbing, but there are many other options including Slip Stitch Ribbing, Emobossed Moss Stitch Rib and Mock Cable Rib to name just a few.
To some extent the selection of ribbing comes down to choosing a pattern you like and that will look good on the project you are planning. You'll also want to choose one that you enjoy knitting. That's why I tend to use 2x2 ribbing fairly often, because I think it's less tedious and goes a little faster than 1x1 rib.
A common complaint that many people have when working ribbing is that the knit stitches on the left side of a column of rib that has two or more stitches in it (like a 2x2 or 3x3 rib, for instance) are often looser than the other stitches.
June Hemmons Hiatt explains this in her book The Principles of Knitting as being because of the distance the yarn has to travel when switching between a knit and a purl. Thus it can be corrected by making that distance shorter, either by putting lots on tension on the yarn when you move to purl or tightening the purl stitch after knitting it, or by working the purl stitches like a combination knitter so less yarn is used to form the stitch. If this is a problem for you, try out these methods and see what works best for you.
Knitting patterns for garments, particularly sweaters, often call for the ribbing to be worked on a needle several sizes smaller than what is used for the rest of the sweater. This keeps the ribbing firm and makes it more elastic (for the most part there's not a big difference in elasticity between the common forms of ribbing).
Going back to Hiatt, she says "you can hardly use a needle too small" when knitting ribbing for a garment. "The more stitches there are packed into every inch of the fabric, the more elasticity it will have and the less likely it is that the ribbing will stretch out and lose its resilience with wear." Good advice.
Twisting Stitches for Elasticity
In Knitting without Tears Elizabeth Zimmermmann suggests knitting ribbing with twisted stitches, that is, knitting and purling into the back of stitches, which she says "yields a certain elasticity" but "is not an essential part of knitting."
Hiatt says working two swatches with the same yarn and needles, however, she found that twisted single rib was wider and less resilient than regular single ribbing. So twist the stitches if you want to for looks, but don't do it to gain elasticity.