What You Should Have
First and foremost, your Android device should be rooted. If you’re not quite sure what that means or how to root your phone, check out our post entitledHow To Root Your Android Phone With SuperOneClick, which actually links to several different methods. Once your phone is rooted, install Titanium Backup. I strongly suggest you also get the PRO Key for the software, which enables it to run without any limitations (and supports the developer for their hard work). It is well worth the modest $6 price tag.
Getting Started
When you first launch Titanium Backup, you should see something like this:
The X next to Dropbox means I did not configure Titanium Backup to sync my backups to Dropbox –