Is it possible to buy a Bitcoin at one exchange, say BTC-e, and then sell it on another, like LocalBitcoins? If it is possible, how would this be done? More specifically, how could you transfer a Bitcoin from one wallet to another wallet that uses a different exchange?
Yep you can! This form of trading is called arbitrage and is not only done with BTC, but with nearly any un-centralized market.
Only drawback here is the financial difficulty of moving fiat from or to Bitcoins, since exchanges don't let you share your fiat balance with others without having to withdraw to a bank account. The withdraw process for most exchanges is known to take up to weeks and in that time BTC may have fallen or rose, similar with deposits. But a few people get away with arbitrage and make a profit in the end of the day.
To send bitcoins from one exchange to another, you simply need to go to your target exchange, and find out what your bitcoin deposit address is.. Once you know that, go to the source exchange and withdraw, or send your bitcoins to the deposit address of your target exchange..
The funds should be over there in about an hour at the most.. Usually they show up much faster than that. I would suggest you try with a small amount first, until you're comfortable with the process.