There are a number of different ways to download apps, as has been mentioned, AppBrain is popular. I still use mostly the Market though. Many are free, though there are often apps that cost money, or a free version of an app, and then a paid for version with more features.
These can be paid for using your google email address, that is linked to your google profile, that will require your credit card/debit card.
Note: any app that you pay for can be returned within 24 hours for a full refund.