Camera Sensor Sizes

The size of the camera sensor can affect
1. the angle of view of the lens (crop factor);
2. noise leve; and
3. dynamic range

Different image sensor size
Medium format - 51x39mm
Full frames (FX) - 36x24mm. e.g.Nikon D3, D700; Canon 1D, 5D

DX format (DX) - 24x16mm for Nikon, 22x15mm for Canon, 21x14mm for Sigma. e.g. Nikon D300, D90; Canon 50D, 450D. Note that Nikon APS-C sensor is the largest of all. :D
Compact cameras - ~8x6mm.

Sensor Sizes

Refer to the charts above, compact camera sensors are about 9 times smaller than the APS-C and 20 times smaller than the full frame format.

In terms of noise performance, the desirable properties of signal-to-noise ratio (SNR) and sensor unity gain both scale with the square root of sensor area. Therefore, typical DX and FX DSLRs will have a SNR that is sqrt(9)= 3 times and sqrt(20)= 4.5 times higher than a compact digital camera respectively.

Because of this, DSLRs can offer high-quality pictures even at high ISO, while compact cameras tend to produce grainy images. Typically, noise starts to be noticable for
FX - @ ISO 3200+
DX - @ ISO 1600+
Compact - @ISO 200+

The noise level is also exacerbated by pixel count where higher number of pixels means a higher pixel/mm2 (higher pixel density) and hence noiser image.

Different sensor types