There is actually only two versions of PS per se. PS and PS Extended.
PS Extended is the same as PS but with a few more advanced features, like 3D capabilities, motion graphics. Both PS and PS Extended are aimed at the professional/advanced user end of the market.
You then have PS Elements and PS Lightroom which are different again.
PS Elements is similar to PS in many ways, with a lot of the features of PS but not all. The controls are generally simpler to use.
PS Lightroom is an editing package as well, again with a reduced amount of editing features but it has a image management package built in. Good if you have or plan to have a huge catalogue of images.
You'll notice the huge difference in price giving you some indication of the actual PS software.
*Student & Teacher versions of the above are available and these are the same full software packages as above, with a huge discount, if you can prove you are eligble to receive such discount.
As Kerioak says, it would help to know what you want it for.