Typically, you have three prices: The "buy price" is the lowest price you can buy a fraction of a Bitcoin for at any particular time. The "sell price" is the highest price you can get for a fraction of a Bitcoin. Any time a buyer and seller agree to exchange Bitcoins for a particular price, an exchange takes place, and a "last price" is established.
So when you have a new exchange, initially you have no price. After a few people place offers to buy or sell Bitcoins, you'll have a "buy price" and a "sell price". As soon as the first exchange takes place, you have a "last price". As people place and remove offers and exchanges take place, these prices update.
You can also track a few other prices over time. The "high price" is the most someone had to pay for a fraction of a Bitcoin over some particular period of time. That means nobody was willing to sell any Bitcoins for less at that time. The "low price" the least someone sold a Bitcoin for over some particular time period. That means nobody was willing to pay them any higher price at that time.