To be short and sweet, Dropbox is a cross-platform file sharing app. It lets you put files in one place, which you can access on your Android device, on your computer, or basically whatever you have. Dropbox syncs the files you have on all your devices, so you should be able to see and even edit some of them when you're using any device with Dropbox installed.
So let's say you're working on an important PDF file. You have to get on the train to get to the office, but you're not finished working yet. That's not a problem. Save that PDF on Dropbox, access the same file on your phone using the app, and continue working on the go. You can pick up right where you left off. I've worked on some documents in the middle of nowhere using my Android device and this app. It is definitely a handy feature to have.
You don't even necessarily have to have the app installed to view your files and folders. You can just go to Dropbox's website and take a look at everything there. You can also share files and folders from the site.
It also works as back up. If you have important files saved to Dropbox and your computer takes a dive, at least you'll still have those important files.
You can share files and folders with different people. These other people can make and save changes to documents. In a sense, this makes Dropbox kind of like Google Drive. At least you don't have to email everything over manually when you save things to Dropbox and share the appropriate files and folders. This can save some people a lot of time.
On Android, you can upload photos right from your camera. You can also protect your files with a password.