I clean the sensor in a tiered system (start with a non-aggressive system and work my way up).
E.g. the easiest thing...
1) Remote the lens. Point the camera at the floor. Invoke your cameras's self-cleaning cycle a few times and see if that does the trick. Often it will.
2) If that doesn't work... grab a hand-squeeze air blower (like a Giottos "Rocket Blower"). Put the camera into manual clean mode (so it locks the mirror up and shutter open). Give the camera a few puffs of air (avoid touching the sensor). That will also usually do the trick.
3) If the dirt is more persistent... then a few swipes with a very soft and pristinely clean brush will usually clean it. I usually give the brush a few hard whacks on the edge of a disk to knock any dust off the bristles (in case anything decided to stick to the brush while it was in storage.) You can even get sensor cleaning brushes that have a grounding wire... so if the reason something is stuck to the sensor is due to static cling... it'll discharge the static.
4) If the dirt still persists... then I go to an aggressive cleaning system.
I use a Sensor Scope to inspect the sensor and see what & where the dirt is located. Keep in mind that everything is actually upside-down and backwards on the sensor. So if you see a dust spot in the upper right corner of your image... then the dust is actually in the lower left corner of the sensor.
I use a web-cleaning system and, in my case, that's "Eclipse" cleaning solution by Photographic Solutions, Inc. (available at most photography stores as well as online) but you also need the correct sized "Sensor Swabs" to go with it. The swabs come in sizes for full-frame, APS-C, and a few others. A few drops (just a few... maybe 3... 4 tops) on the swab. Then do a gentle wipe from one edge of the sensor to the other (do not rub back and forth ... just a single swipe should do it). Immediately toss that swab in the trash (do not risk a putting a scratch on the sensor to try to save a swab that costs less than a couple dollars.) If the dirt is still there, I grab a fresh swab.
You can test the sensor by putting the lens on, dial in a high f-stop (e.g. f/22) and take a photo of a plain white wall (you don't even need to focus... in fact it's almost better if you deliberately do not focus the lens.)
If there is still dirt/dust on the sensor, you should see it in the test photo.
There are other cleaning products. For example, I somewhat recently came across a cleaning system that uses some sort of gummy material on the end of a stick. You press the gummy material to the sensor and pull it away and the dirt is supposed to stick to the material instead of the sensor.