The Basics: Keep Your Phone Locked
It's extremely easy and effective to lock down access to your Android phone using built-in tools. Simply tap your way to the "Location and Security" setting. There you can create either a standard password or a screen lock pattern. If you use the screen lock pattern tool, be sure not to make an extremely easy and obvious pattern.
Avoiding Malicious Apps
Viruses and other malware are increasingly targeting the Android platform, so it's important to take a few precautions to avoid infecting your own phone or tablet.
For the ultimate in security, I recommend only downloading apps from the official Google Play store, where they have been vetted. Scammers have been known to create fake versions of popular apps, which are loaded with malware and only available at third-party websites.
However, some apps may simply not be available on the app store. If you see an intriguing app, you can verify its reputation by searching for it on Google and seeing if it has been written about, positively or negatively. Be wary of totally obscure apps.
Note App Permissions
Take the time to learn what permissions an app wishes to access, from making phone calls to using your system tools. While it makes sense that a social networking app may wish to access your contacts, it doesn't make sense that a "dancing dog" app would. Be wary when apps ask for more than seems to make sense.
Consider an Anti-Virus
Finally, you can consider an anti-virus program for Android. Savvy users may not need these programs, but apps like Lookout Security & Antivirus do offer powerful features for added protection.
Find Your Lost Phone and Erase Data Remotely
Thanks to GPS and wireless Internet, you can locate a missing Android phone using one of the several apps available. I recommend Android Lost (free). Even with your speakers turned off, you can use this app to play a sound, which is handy when you simply lose the phone in your house. Furthermore, this app also enables you to erase all the phone's data, saving your contact information and files from prying eyes.
You can also take pictures from your camera to know who is using your phone. And better yet: This app can be installed remotely, so you can install it after you have lost your phone. (However, I strongly recommend installing it right away.)
And Don't Forget Physical Mishaps
All the security measures in the world won't matter if your phone breaks from a simple drop. Be sure to invest in a quality case.