When you apply for a car loan—
—they will pull your credit report and perform a "hard inquiry" on your file. This means the inquiry will be noted in your credit report and count against you, slightly. This is perfectly normal. Just don't apply too many times too soon or it can begin to add up. They will want proof of your income by asking for recent pay stubs. With this information, your income and your credit profile, they will determine the maximum amount of credit they will lend you and at what interest rate. The better your credit profile, the more money they can lend and the lower the rate.
If you know the price—
—that you want financed (the price of the car minus your down payment) that is the amount you can apply for and in that case the only factors they will determine are 1) whether or not you will be approved and 2) at what interest rate you will be approved. While interest rates generally follow the direction of the prime rate as dictated by the federal reserve, there are market fluctuations and variances from one lending institution to the next. Further, different institutions will have different criteria in terms of the amount of credit they deem you worthy of.
So in your case—
—you know the price of the car. Now determine how much you want to put down and take the difference to a bank or credit union. Or, work directly with the dealer. Dealers often give special deals if you finance through them. A common scenario is: 1) A person goes to the car dealer 2) test drives 3) negotiates the purchase price 4) the salesman works the numbers to determine your monthly payment through their own bank. Pay attention during that last process. This is also where they can gain leverage in the deal and make money through the interest rate by offering longer loan terms to maximize their returns on your loan. It's not necessarily a bad thing, it's just how they have to make their money in the deal. It's good to know so you can form your own analysis of the deal and make sure they don't completely bankrupt you.
The most important thing—
—is that you can comfortable afford your monthly payment. The car dealers don't really know how much you can afford. They will try to determine to the best they can but only you really know. Don't take more than you can afford. be conservative about it. For example: Think you can only afford $300 a month? Budget it even lower and make yourself only afford $225 a month.