Can you represent an audio signal in any other domains besides time and frequency?
Basically it's all about breaking down the information into several 'bits'.  The actual audio signal is a time varying 'value', however often it is useful to consider it in a different form.  As an analogy, consider the number 256:  depending on what you're doing with your numbers, it might be useful to treat the number as 200 + 50 + 6, or 16 + 240, or 16*16, or maybe as 2^8; there are an infinite number of ways to treat the number, and which one you use depends on what you're trying to achieve.

The 'frequency domain representation' is an example of doing the above break-down, but with a signal rather than a number.  In this case, you're representing the original signal as a sum of sinusoids, all with different frequencies, amplitudes, and phases.  If you add them all together, you get your original signal back.  You could alternatively choose to represent it in a different way, such as with wavelets, or any other approach that may or may not even have a name (yet), if it is useful for what you're doing to it.  Maybe you could break it into 3 second chunks, then rearrange the signal in each of those chunks to be monotonically increasing, and remember how you reordered them.  This sounds a bit ridiculous, but there is an approach to manipulating MRI images with this sort of reordering (it doesn't use the chunking part, but that's just an example of how you can do whatever you like).

One advantage of sinusoids is that, like Nathan Day says, it relates to the way our ears interpret the pitch of sounds.  But a more important reason is that sinusoids are complex exponentials, which are eigenfunctions of linear systems; that is to say that linear systems are much much simpler to analyse if you consider the inputs and outputs as sums of sinusoids.  That is the main reason why Fourier analysis is so widespread and important.

Short answer to your question: there are an infinite number of domains you can represent an audio signal in.  For another popular one, see wavelets.