The term full frame sensor means that the sensor size is 36mm x 24mm. This is the old 35mm film negative size and has become the standard for top quality digital images.
Why is this important? If you have a full frame sensor it means it is bigger than other sensors and can therefore hold more detailed infomation. Most consumer DSLRs or Mirroless cameras have smaller sensors which cannot hold the same amount of information for the image. This is one consideration in sensor size.
The main consideration in terms of sensor size is what effect does the sensor size have on the focal length of my lens? What does this mean? As most lenses are designed for a full frame 36 x 24mm sensor, the focal length on your lenses will be true if you are using one. eg if you are using a 24 - 70mm lens on a full frame camera, you are getting a true 24 - 70mm. If you have a camera with a smaller sensor, this acts like zooming in on your image in terms of focal length. Its called the Mutiplier Effect. If you are using a consumer DSLR, it normally means the focal length is multiplied by a factor of 1.5. This will change the 24 - 70mm lens into a 36 - 105mm lens. Multiplied by a factor of 1.5. If you are using a Mirrorless camera, the multiplyer factor can be as high as 2. This means the 14 - 42mm lens on the Olympus OMD now becomes a 28 - 84mm lens.
Why is this important? The focal length of your lens will determine the perspective of your subject. If for example, you want to do a really great portrait of a subject, you should use a focal length of between 80 - 90mm. If you are operating a system with a mutlplyer factor of 2, it will mean you should be choosing a focal length of 40 - 45mm to get the correct perspective of your subject.
To find out your multiplyer effect ratio, have a look in your camera manual and see the section on lens. It will give the focal length for your lens and then say the 35mm equivalent. This will tell you the mutliplier to use with your lens to get the actual focal length you are using. An example of this might be. Your camera manual might say your lens is an 18 - 55mm, or (27 - 83mm, 35mm equivalent).
Cameras with smaller sensor size can still give you incredible image quality, but you need to be aware of the Mutlplier Effect and how that will change your image perspective.