What is the difference between a full frame sensor and a crop frame sensor?

Why is full frame so expensive, what are the advantages that full frame can do, actually I should ask the other way around, what some advantages of crop frame sensor, lets say nikon D300 vs D700.

Basically the larger the sensor the higher quality the image. That should be clear. Full frame sensors typically have less noise in the image file than smaller sensors

The advantage to having a crop frame sensor is obvious if you shoot sports.

With the D3s (D700) using a 300 mm f/2.8 lens, I can shoot from the sidelines, moving with each play.

With the D300 using a 300 mm f/2.8 lens, I get the equivalent focal length of a 450 mm f/2.8 so I can be further away and still get good shots with a minimum of movement

So, using a cropped sensor camera, I can reach further without loss of light I would get if I added a telextender 1.4x (one stop).

It is for this reason, most of us have one cropped sensor camera and one full frame camera

Sample of a cropped sensor camera on a long lens

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NOTE: Read Hondo's answer. He covers some other important aspects when it comes to sensor size.

Crop frame just means the sensor is smaller than the full frame. The most expensive part of a camera to manufacture is the sensor. A larger sensor means a more expensive camera. In addition, the only people who really need full frame are usually professional photographers, and pro-level cameras have other more expensive bells and whistles.

In terms of image quality, full frame general has somewhat better low light performance as the pixels on the sensor are larger than on a crop frame. In other words, a 10MP full frame sensor will have larger pixels than a 10MP crop sensor. In addition, it is possible to get a shallower depth of field with a full frame sensor. Other than that, there's not much noticeable difference.

If you are a professional photographer, you might be able to tell the difference between a photo taken with a full frame vs.a crop frame camera. If you are not a professional, a full frame camera is really not worth the extra expense.

I really can't say with any certainty but I have read many of the answers given by 'fhotoace" and "Jim A." and they both ex told the virtues of a full frame sensor like those found in Canon & Nikon DSLR cameras.
That said, it is still possible to get needle sharp photo's with "off beat" cameras. I have made 11x14's and 16x20's which are as sharp as I would ever want them. (And I'm pretty fussy). With the basics of using maximum pixel, a small f/stop f/11 and smaller f/stops, and ISO's of 200 or less. If you are in the place in life of getting your first DSLR, I would wholeheartedly recommend the full frame sensor. If you have enough "smarts to want to know the difference, it certainly makes sense to "go for the gold" in a matter of speaking and get a camera with a full frame sensor.

Size difference. Bigger sensor, more details captured. The only advantage of a crop sensor as I see it is lower price.