A DSLR's sensor has an electronic charge and just like a tube television, it attracts dust. Changing lenses, zooming, even something as minute as focusing can bring dust into the camera. The best way to determine exactly how much dust is on your sensor is to take a photo of the sky at f16 or higher.
Also make sure your camera is set to manual focus - otherwise the lens is going to keep seeking for a focus point. Next, load up your image in photoshop and enlarge it to 100%. That'll give you the ability to see every tiny bit of dust that's plaguing your sensor.
So now that you see the dust, what do you do with it? There's always the option of cloning out all the dust in photoshop, but that's an hour's worth of time that's better spent behind the camera. If you have the courage you can clean the sensor yourself with the right equipment and supplies.
If you decide to try it yourself, check to see if your camera has a "clean sensor" option under the set-up menu. Many DSLR's require you to have a fully charged battery in order to clean the sensor. If the "clean sensor" option is greyed out, that is most likely the case.
Charge your battery fully and give it another go. If you have an older SLR, it is possible that it does not have a "clean sensor" option. At that point, in order to access the sensor yourself, your only option is to put the shutter on bulb and clean it that way. It's a way to do it but it's very risky. If you accidentally release that shutter while cleaning it's going to be a costly repair. Also the sensor will be activated, which will increase the chances of more dust appearing.
Whatever you do, DO NOT use compressed air to clean your sensor. Many types of compressed air emit tiny particles of liquid propellant which can damage your sensor.
Always use products meant specifically for sensor cleaning, such as Sensor Swabs and E2 solution by a company called Photographic Solutions. Bulb blowers are a great way to do a quick cleaning - never do a wet cleaning unless it is absolutely necessary. You can even buy small, travel-friendly bulb blowers for cleaning on the go.
There's nothing wrong with wanting to try it yourself, but be aware - Some camera manufacterer's warranties become void if you clean the sensor yourself.