When you talk, sound is produced by passing air over our vocal cords (flaps of tissue in our throats) causing them to vibrate. The vibrating vocal cords make the air around them vibrate and this makes sound. We can change our voice by changing the shape of our throat and mouth, allowing us to make different sounds for different words. Our voice comes out in the form of waves, and it is the frequency of these waves that makes a high note sound high and a low note sound low.
But our voices aren't quite this simple. You know how different instruments sounds different even if they're playing the same note? That's down to 'timbre'. Our voices have a timbre too - they are not just one pure note but a mixture of lots of different waves of different frequencies. When air moves through our throats, it vibrates and you end up with lots of resonance frequencies, or harmonics, that sound different between different people. Someone can sound all squeaky at the same pitch as someone else who sounds normal depending on the harmonics of their voice (explaining why some people sound great on that TV singing shows and others, well, they just don't).
Now the fun bit is when we start thinking about what happens if we breathe helium. Sound waves behave differently depending on what they are trying to move through. Normally, we breathe air, which is a mixture of gases containing mainly nitrogen and some oxygen. Sound waves generated by our vocal cords move through air at a constant speed of around 350 meters per second.
But helium is less dense than the air we normally breath and this means that sound waves travel through it much faster than usual (around 900 meters per second) during the short time that they are making their way through the throat and out of the mouth. The pitch of our voice isn't actually altered, but the timbre or quality of the voice is. Basically, the fast moving higher frequency sounds have more power as they float about in your throat, and the lower frequencies get a bit lost. So our voices end up sounding flat and you talk something like Donald Duck.
I know a lot of people say that your vocal cords vibrate faster in helium, but there is no evidence for this. There is also nothing to back up the idea that the pitch of our voice is higher after breathing helium (although you'll probably automatically talk in a high pitched voice if you breathe helium because you expect your voice to go higher). If you don't believe that the pitch is really not altered, try singing before and after breathing helium (you'll be able to hear the pitch better). You should find that the sound your voice makes isn't actually higher, just squeaky and flatter.
So if a lighter gas like helium gives us a squeaky voice, wouldn't a denser gas like Argon or Xenon amplify the lower frequencies and make it richer? This is actually true, but the problem is that these gases are harder to get hold of than helium. Xenon for example is very expensive, and in addition, it has anesthetic properties similar to nitrous oxide, so you can get a little high if you breathe it. Just take my word for it and don't try it for yourself.
Something to remember: Although helium is un-reactive, it can still kill you. If the entire room was filled up with helium, you would quickly asphyxiate due to there being no oxygen present. Please do remember this if you insist on breathing in helium - while a balloon-full isn't going to do you any serious harm, that lightheaded feeling you get after breathing in too much balloon gas is a sign that you're not getting enough oxygen into the lungs and you should probably give the balloon breathing a rest.