Both of these refer to interconnections between the pre-amp and the amplifier, and less commonly, between the source (CD player for example) and pre-amp.
Single-ended is, I believe, the same as "unbalanced" -- the usual interconnect method -- the one with standard "RCA" jacks at each end. It has only two conductors, and is not protected against outside electrical interference or "noise". You're already familiar with this one.
Balanced connections are less commonly used, though quite a lot of "high-end" gear supports them. The connectors are large, round 5-pin "DIN" connectors, and have 2 more conductors (counting the shield/ground) than unbalanced connections. These are known as "microphone" connectors because they are used for that purpose in recording studios. Their main claim to fame is that they are protected against electrical interference, and they work well for long runs.
Most agree that balanced connections are the best choice if you must use long interconnects. The verdict on shorter runs is less clear. Some (perhaps most)people prefer unbalanced connections where possible (the electronics to support unbalanced connections are simpler and therefore give a more direct signal path). Some, though, prefer balanced connections everywhere (I would probably be in this latter camp if my sources supported balanced connections -- I'm not able to keep my power and signal cables as well-segregated as I'd like)
For myself, I was relieved to get very good results with homemade balanced interconnects for my 40' run between preamp and apm. I think it's safe to say that the sound difference on my system is minimal between the two types. I followed directions and parts lists posted here, and which I am happy to re-post if you want them.
One issue with balanced connections is that some equipment supports the connection, but the electronics behind it isn't "truly balanced". This is the case with my Classe CP-35 pre-amp, I'm told, but thankfully I don't hear any degradation. I dont' understand the electronics well enough to explain this issue.
For most people, support for balanced connections is not a deciding factor in equipment purchases unless they have unusual cabling requirements.
I suggest you do some searching here on "balanced unbalanced" and you'll find a lot more.