There are two parts of your blog that you should back up. The first is your blog's content. You know all of those posts you've written? What would happen if they all disappeared?? What if you want to move them over to a new blog? Did you know you can export your blog? This is something I do monthly for mine.
In your blogger control panel, look for "Settings" over on the left. Then click "other" at the bottom. At the top of your new screen, click "Export blog" and save the file somewhere safe on your computer. That's it!
The other part of your blog that you'll want to back up is your blog's design. This is important because you don't want to have to start all over if you make a mistake when adding a gadget, trying new html, etc. It's especially important if you have a custom design and you aren't quite sure how to put it back yourself. =)
In your blogger control panel, click "Template" over on the left. Then look in the top right corner and click "Backup/Restore" so a new window will pop up. In the new window, click the orange "Download Full Template" button and save the file somewhere safe on your computer. Easy as that! If you need to do an emergency restore of your blog's design, you can go right back to this same place and upload the file you saved. I do this each time I tweak my blog's design.
Now you just need to remember to back up your blog regularly. I sent monthly reminders for mine because we all know how easy it is to forget. I hope that was helpful! If you're looking for a new blog design, Design by Christi just might be what you need. =)