The "price" is usually based on the most recent trade in the specified exchange pair (e.g. LTC BTC). The most recent trade is simply two other participants on the exchange, the seller and the buyer. The seller (also called a "maker") offered to sell a particular coin at a particular price in a particular currency (e.g. to sell some amount of LTC for a specific price in BTC). The buyer (also called a "taker") offered to buy a particular coin at a particular price in a particular currency. The trade happens by the exchange when a buyer and seller's orders are matched.

Can you make money in these price differences between multiple currencies? It is possible, this is called an arbitrage. However, don't forget that the transfers usually take a bit of time and the arbitrage opportunity may have passed by the time you've made some trades, because rest assured others are looking for these arbitrage opportunities as well. Additionally, there are usually fees for each trade that eat into any earnings you might be making.

Also, this isn't that different from Forex Arbitrage trading in fiat currency markets so it is a well studied discipline and we can certainly learn from the practices established there.