If your home has an active sump pump, be warned: your sump pump will not last forever. I promise. It probably won't give you any warning before it fails, either. It'll be a silent event that happens in the middle of the night during a heavy rainstorm. You'll know it failed when you awaken to a flooded basement.
I've had many neighbors with sump pump problems and I've heard from countless home inspection customers who have had problems in the past. Having a flooded basement due to a failed sump pump almost seems like a rite of passage for homeowners, but most would prefer to remain uninitiated. If you're in that crowd, install a backup system. This is really important stuff, and it's one of those things that most people don't think about until it's too late.
There are two common backup systems; battery and water powered.
Battery-powered pump
The most common type of backup sump pump system is a battery-powered system. This consists of a big heavy battery that's about the size and shape of a car battery, as well as a backup sump pump that sits in the sump basket a little bit higher than the primary pump. This system will save your basement from flooding if your sump pump fails or your power goes out.

The Basement Watchdog is a brand that makes backup batteries along with a warning system to tell you if the battery has a problem. This is a good feature to have because I find that many backup batteries are dead.
Lowes has a nice video showing the steps needed to install a backup sump pump, which I've embedded below. The process is pretty simple, but if you're not comfortable with any of this stuff, hire a plumber.
Water-powered pump
Another backup option is to install a pump that is powered by the municipal water coming into your home. The nice thing about hydraulic sump pumps is that you don't have to worry about keeping a battery charged all the time, and if you have an extended power outage, you won't have to worry about the pump failing.
The downside to using a hydraulic sump pump is that they use a lot of water and they're slow. The Basepump and the SumpJet both use 1 gallon of city water to remove 2 gallons of sump basket water. The Burcam Sump Buddy, which I installed at my own house, seems to use a lot more than that. I can't tell from their technical data exactly how much, however.