Simply, crop factor is the ratio of sensor size to a reference 35mm (or full frame) sensor which measures 36mm x 24mm.
|By MarcusGR [CC BY-SA 3.0 (http://creativecommons.org/licenses/by-sa/3.0)], via Wikimedia Commons|
There is a bit of math that goes with this but it's not that interesting really so please just look at the picture.
Why is any of this important?
Well, back in the day when 35mm film was a very popular format everyone knew what to expect from a particular focal length of lens - i.e. they knew what to expect in terms of field of view for that lens. This made discussions about what lens to use for a particular type of photography pretty easy as everyone was singing from the same hymn sheet, so to speak.
Then digital sensors hit the scene but were, initially, for economic reasons, smaller than 35mm and this brought its own set of issues. How so?
A lens is circular and a sensor is rectangular. The image projected by the lens is, very originally, known as the image circle and this lands on the sensor - a full frame sensor is one that covers the full area of the image circle whereas a crop sensor is one which covers a smaller area of the image circle. In effect the rest of the image is thrown away or 'cropped'.
By Self En:User:Ravedave (Self En:User:Ravedave) [GFDL (http://www.gnu.org/copyleft/fdl.html), CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0/) or CC BY 2.5 (http://creativecommons.org/licenses/by/2.5)], via Wikimedia Commons
In the picture above, the full frame sensor is in red and the crop sensor is in blue and it can be seen that the crop sensor looks more zoomed in - in other words it has a narrower field of view than the full frame (assuming each version is viewed at the same relative size).
And this is where the confusion begins. A lens on the full frame will render an image differently to a crop sensor even though the lens' focal length remains the same. So going back to the beginning, I said that crop factor was a ratio of sensor size compared to full frame - this means there is more math that I won't bore you with which ultimately leads to a thing called equivalent focal length.
A Canon APS-C (crop sensor) has a crop factor of 1.6 (take my word for it otherwise I have to show you some math!). This means that a 50mm lens on a camera with that sensor has an equivalent focal length of 50 x 1.6 = 80mm as compared to a full frame sensor. Now you can see why it can start getting confusing as to what kind of image you will get with a lens as it will vary dependant on sensor size.
Crop factor cameras are also cheaper to produce and they can have there own lenses too. These lenses will use less glass due to the decreased image circle and are therefore cheaper than full frame equivalents. Another advantage of a crop factor sensor is that apparent 'zooming' effect which can be good for wildlife and sports photography as it allows the use of a cheaper telephoto e.g. a 400mm lens on the Canon crop sensor above will be the equivalent of 400 x 1.6 = 640mm. It would cost an awful lot more to have a Canon 600mm lens than a 400mm!
Like all good things there are a couple of downsides - one of which is that, generally, there is more digital noise from a crop sensor than a full frame one. Technology is moving so quickly that this is becoming less of a problem with every new release of a camera.
So there you have it; a very brief foray into sensor sizes and what it can mean for your photography.