Android Browser was phased out of the most recent version of Android. If you have Android 4.4 KitKat, you don't have Android Browser and you're safe from this flaw. You can find out which version of Android you're using by going to Settings>About Phone. Underneath the header "Android Version" it will tell you which version you have. If the number is higher than 4.4, you're fine.
If it's lower, you still have Android Browser on your phone, although you might not be using it. If you don't use Browser, you still might want to disable it just to be sure. Go to Settings>Apps>All. Find Browser. It has an icon that looks like a blue globe. Press it and then press the disable button.
After you've disabled Browser, go to the Google Play Store and download the browser of your choice. Chrome and Firefox are fast and secure. You can't go wrong with either of them.