My favorite backup software is SyncBack.  There is a paid and free version, but I find the free version to be more than sufficient for most purposes.  You can download it by clicking here.

Of course, once you have backup software, you need to have some place to store your backup.  The most secure and safe backup is an off-site backup (discussed below), but if you just want to backup locally in your home or place of work, then a USB or NAS (Network Attached Storage) solution is your answer.  If you just have one computer, then a USB external drive is all you need.  It plugs in and your computer will assign it a drive letter.  It looks and acts just like your hard drive inside your computer.

If you have more than one computer, then a better solution may be to purchase a NAS drive.  This box looks just like a box that is USB, but it will have a network port in the back, and you connect it to your router instead of your computer.  Most of them come with software to simplify connecting to the NAS.  The NAS will look like another computer on your network, and you can backup as many systems as you like to them.  Some are Window or MAC only, so be sure to read the specifications before you buy.  Currently I like the Buffalo Linkstations because they are simple to set up, inexpensive and work with both Windows and MAC computers.

The best backup solution is one that stores your data "off-site", or in other words, away from where the original data is stored.  In the event of a fire or burglary, you don't want to lose your original AND backups.  I recommend using Carbonite to do off-site backups.  The great thing about Carbonite is if you can turn on your computer, then your backup gets done.  You can get their software directly from them, but if you contact me, for the same price I'll install it for you and give you tips you won't get from Carbonite.