Chrome: While Dropbox has clients for operating systems, there is nothing for the browser apart from the website itself. QuickDrop is a free Chrome extension that gives you easy access to your Dropbox, and speeds up downloads and uploads.
Install it, give it access to your Dropbox, and a click on the extension icon will show your cloud storage in a drop-down pane. You can download, rename and delete any file, as well as get a link to share it. There's no option to move it though.
In QuickDrop's settings, you can also point it to its own folder (you'll have to manually create that in Dropbox) for quick uploads. When you see an image or file on the web that you want to save, right-click it and choose "Download to Dropbox" for it to go straight to that folder. To upload a file from your hard drive, drag-and-drop it to a new tab, where you will see an option to automatically upload it to that folder.
There are some cool Dropbox extensions that do these tasks already, but QuickDrop puts them all in one place. Thanks Daniel!
QuickDrop [Chrome Web Store]