You download the apps directly on your phone. It is possible to download them to your PC, transfer, and install. But, that requires you use your phone to download an app to do that. Soooooo, you're better off just using the Market on the device itself.
The following is based on the HTC Droid Eris Sense UI. Your labels may vary
When you are in the market, and you find an app you are interested in, tap on it. You should get a brief description of the app. At the bottom of that first page should be a button labeld "Install" (free apps) or "Buy". Tap that.
Now you should be on a screen that lets you know what the App will have accesss to (i.e. Network Comms, System Tools). The options on that screen should be "OK" and "Cancel". Hit OK.
The phone goes back to the Mrket, but the Notification Bar should show that the file is downloading, installiing, then finished. You can then go to your Program Menu and launch it.
Hope that helps...