The Files tab in the web interface will bring you to your Dropbox folder. This folder is exactly the same as the folder on your devices; any files that you've placed into your folder on your computers are accessible through this part of the web interface.
When you click on a file in the web interface your browser will make an effort to open it. This is simple if the file in question is a common text or image file, but your success with more complex files will obviously be limited.
You can download files using two different methods. If you simply want to download a single file you need to hover your mouse over the file in question. The file will become highlighted and a drop-down menu will appear on the right. If you click that arrow a list of options will appear, one of which is the Download File option. It does what you'd expect. You can also download an entire folder using this method: folders are retrieved as .zip files