Is it possible to use a dropbox file in an online application? For example, I am applying for a job using my mobile browser and they ask for my resume. If I have my resume on my dropbox folder, it seems like there should be some way I can "open" it to be uploaded for the application process...
That would probably depend on the mobile browser you're using and whether or not they support opening files from Dropbox. It would be up to the browser developer to implement such a feature using the Dropbox API.
The best way to access the files in your Dropbox account from your mobile device is to install the Dropbox application on your mobile device directly from the App Store.
The Dropbox app will give you a "window" into your account. You will be able to view all the files and photos in your account (the app is password protected) and you will be able to download the files you want directly to your phone.