Signal in the wire of headphones
Usually audio frequency electrical signals flow through headphones cables. These get converted from an electrical current/voltage to acoustic vibrations by the speaker.

I say "usually" as it is possible to embed things like play/pause/answer etc. on to the cables - usually though this is only done on the microphone cable of the headphones not the speakers.

A standard 3.5mm headphone jack on something like a phone or media player will output an analogue signal which can drive a load of maybe a few 10's of mW (see this EE.SE question). That is pretty much enough to drive some small speakers as are found in headphones. Computers or laptop line-out ports will likely be able to supply a more power than that.

In terms of standards, I'm not aware that there are any (someone correct me if that is wrong). There are nominal voltage levels which range from \$500\mathrm{mV_{pk}}\$ to as much as \$2\mathrm{V_{pk}}\$, and they can be AC coupled, but beyond that will depend on the manufacturer.

Headphones cables are usually narrow gauge (low power rating!) with minimal if any shielding from noise, crosstalk or anything else. This is done to ensure they are light weight (the last thing you want is a heavy copper cable hanging off your ears!). How much noise this results in, who knows. Professional grade headphones will probably be designed with better shielded cables or whatnot, but the consumer graphe 'phones you buy for a few £/$ are probably not designed with noise in mind.

As for wavelength, well, a \$4\mathrm{kHz}\$ signal has an electrical wavelength of about \$50\mathrm{km}\$, so it's not really a concern :)