The drive axle on your car is just that – an axle that drives your car along the road. On the earliest vehicles, was a solid rod with two wheels attached, turned by a chain; just like on a go-kart. In your car or truck the drive axles still perform that same function, delivering power from the engine (via the transmission) to the wheels, and then to the road, but it's now more complex than just a solid rod. Cars and trucks with a solid rear axle rely on two rods inside the differential to deliver power; this allows the wheels to turn at different speeds while cornering. But modern front-wheel-drive cars (or rear-drive cars with independent rear suspension) have much more complex axles with a constant velocity (CV) joint on either end, which allow the wheels to steer and follow bumps in the road.
Modern drive axles are typically short rods with a flexible rubber boot covering a CV joint at each end. They are also commonly referred to as "half shafts". The boot is important as it holds the grease inside and keeps dirt out, and the grease is important to keep the CV joint functioning properly. This increased complexity means increased chance of something going wrong, but in return modern vehicles have better traction, ride, and handling.
CV joints are tough. In normal use, they'll usually handle years and more than 100,000 of miles of bumps and corners without trouble. But if the rubber boot surrounding them should get torn it will allow dirt and water to get in and grease to leak out, which will cause the joint to quickly go bad. Just a small amount of wear inside the joint allows play between the parts; listen out for a clicking noise that gets noticeably louder when turning a corner. If left alone it is just a matter of time before the joint breaks and you're left without drive.