Raster images are images represented by pixels, or dots. The number of pixels is fixed, so as you blow up the image you also blow up the pixels, losing quality. 99% of what Photoshop does is raster.

Vector images are images represented by lines and curves that are mathematical equations. As you blow up a vector image, the quality never decreases because it's all mathematical, and the imaging software just interprets and redraws it at an increased resolution. Adobe Illustrator is a program used to produce vector works.

If you are working in photoshop, the logo will likely be raster. Just make sure you make it large enough that if he has to print it big, it will look ok. If he wants a vector version of the logo, I suggest making it in Illustrator.

The benefit of raster is the level of control, you can do a lot more with photoshop and you can do it a lot more easily. Also the spectrum of results you can achieve with photoshop are a lot greater.

The benefit of vector is of course quality. No matter what scale you print at, the quality is always perfect because it's all mathematical. But vectors are harder to work with in my opinion ...

See the wiki links below for more info on raster vs vector.