Cat 5e is cat 5 with a slightly improved design that should result in less crosstalk (electromagnetic interference). If you just have a 10Mbps network (unlikely, unless you have an old router/switch/hub), it won't make a difference, but Cat 5e might perform better if you have a 100 Mbps network. Additionally, if you have gig (1000 Mbps), I don't even think the original Cat 5 can handle it, but Cat 5e can. Basically, between the two, Cat 5e is your better bet. (I think it's actually difficult to find original Cat 5 cable these days, but I could be wrong.)
Cat 6 is going to be more expensive and probably not worth it unless you are laying non-easily-replaceable wire, say, through your house (or unless you don't find the cost difference that significant). It might handle gig a bit better than 5e, but both are perfectly acceptable; cat 6 (unlike 5e) is expected to work with future 10 Gbps networks, but I assure you that will not matter, especially at your house, for some time to come.

As for the differences, it's basically the same cable, but it improves the speficiation to further reduce crosstalk and noise--but all three of these cables are made with the same number of wires, same materials, and are backwards compatible.
Basically, your best bet is probably 5e.