Every digital camera has at its heart a chip, called a sensor, that registers the light captured by the lens. The size of this sensor varies with the camera and relates back to old 35mm film days.
A full frame (FX) digital camera has a sensor measuring 36x24mm i.e. 864 sq. mm.
A cropped format (APS-C) camera has a sensor size of 23.6x15.6mm (Nikon) i.e. 368 sq mm or 22.2x14.8mm (Canon) i.e. 328 sq mm.
Comapct cameras have various sensor sizes but a reasonale quality camera size would be 8.8x6.6mm i.e. 58 sq mm.
A Fuji bridge camera sensor is typically 6.4x4.8mm i.e. 30.7 sq mm.
A quick glance above shows that a full frame sensor is over twice as large as a cropped frame sensor and that other sensors are tiny in comparison. So you should buy a full frame camera for best quality images, right? Well, yes and no. The size of the sensor must also be combined with other factors such as the number of megapixels on the sensor, angle of view of the lens used, and filters and software algorithms used to clean and condense the image.
The more megapixels you have on a sensor, the better the image in theory. However there is a balance between the number of pixels on a sensor and the spill-over of light between those pixels. Too many pixels creates distortion of the image resulting in "noise".
The lens and sensor combination are also important. A 100mm lens on a FX camera witll give a certain zixed image. If a 100mm lens is used on a smaller APS-C sensor the same image covers more of the sensor giving an equivalent magnification factor or 1.5. This means only the sharper central portion of the lens image is captured thus improving quality.
For the most part full frame (FX) camera use less megapixels than APS-C cameras to give a good image. However, current high-megapixel APS-C cameras probably produce equal quality images when viewed at lower magnification.
It is only at really high magnification for large prints that the noise from all those tiny megapixels becomes obvious.
So what does all this mean? Well with a smartphone, compact or bridge camera, you are never going to get the quality of a DSLR. The small sensor size combined with a small lens just gives average results.
Professional photographers tend to use FX cameras for several reasons other than sensor quality. They need the faster professional lenses and the rugged build quality that comes with FX cameras. Professional landscape photographers use even larger format cameras to ensure maximum print quality.
However, if you are just an ordinary keen amateur like myself then a good APS-C (Nikon DX format) camera is more than adequate for all but the most discerning photographer. And the camera is only part of the story... a good camera requires a good lens. But the best sensor is the photographer's eye for a photo.