When getting into the technicality of the cuts, this can affect the nutrition and taste of the hay. For example, first cut timothy generally will be a more stalky hay with thick stems, that is lower in protein, second cut timothy will be a leafier hay higher in protein. It varies a lot for each individual type of hay, how long the grass/legume is allowed to grow before it is cut, as well as where the hay was grown. Most rabbit owners don't need to worry about the different cuttings too much, unless you have a very picky rabbit that will only eat a particular type of hay, or you have a rabbit with digestive problems that needs a particular type of hay. But for most people and their rabbits, you just want to shoot for a good green grass hay that has a mix of leaf and stem.
http://www.rabbit.org/journal/4-7/hay.html
A good basic all around type of hay for the average bunny owner is just the bags of timothy you would get from the pet stores. I believe that most of that is a second cut timothy. It is usually a good green color with a mix of leafy parts and stems, which you want both of for good nutrition and digestive function for your rabbit. The leafy parts will provide more protein and nutrients, and the stems provide indigestible fiber which rabbits need for good gut function. You can also get the other types of grass hay and do a mix.
Pet store hay is expensive so if you are wanting to buy cheaper bales of hay, then you are still wanting to find something like the pet store timothy hay if you can(a mix of leaf and small stems but doesn't have to be timothy). But otherwise most types of good horse quality(clean, NO mold, and preferably weed free) grass hay is just fine for rabbits.
Some places will recommend feeding alfalfa hay to young rabbits, but I wouldn't recommend feeding it exclusively as it can be too rich of a diet with too much protein, and it can make for picky buns because rabbits tend to really like the taste of alfalfa. If you are feeding unlimited or close to unlimited, alfalfa based pellets to a young rabbit, then feeding grass hay is fine. They will get the nutrients they need from the pellets, and fiber from the hay. Feeding a little alfalfa hay in addition to the grass hay, would be ok.