The Dropbox app for iPhone and iPad has been updated recently. There's a UI redesign in there now that takes full advantage of the Retina Display, so your shared files will look better than ever. The iPad version has gained a landscape view, and the app will now finish uploads and downloads in the background. Dropbox also allows file caching now, so once you've viewed a file, you won't have to go back and in and re-download it again. I presume that means only while the app is running -- if you have to actually close the app for any reason, you might have to connect up to see the file again. [Update: No, cached files remain accessible even without network access. –Ed.] But you can find out for yourself because the Dropbox app is free (as long as you sign up for a free account with their excellent service).
The company has also released something called the App Directory, which is an online database of iPhone and iPad apps that will work in conjunction with Dropbox and its file sharing capability. You can browse, review, and rate Dropbox compatible apps (and actually, Android, Blackberry, and Windows Mobile apps are included as well, if you happen to have one of those phones), and even follow links straight from the listings to download and use whatever apps you need. If you save a lot of things on your Dropbox account, you might want to take a look through there. There's standard stuff like GoodReader and Documents to Go, but odds are that, even if you've found another use for sharing on Dropbox, there's probably an app to help you do it.
Note: Please don't put Dropbox referral links in the comments. They will be deleted, and repeat offenders will be banned.