First a quick definition. A full frame digital camera has a sensor the size of a negative from 35mm film camera days. The sensor is 35.5 by 24 mm.
An APS-C digital camera has a sensor that is smaller (about 1/3 smaller). In this camera, the sensor is 23.5 by 15.6 mm.
So which is better?
Well it depends on your definition of "better". Let me explain.
If cost is your driver, the APS-C cameras tend to be less expensive. Often, they are quite a bit less expensive.
If image quality is your driver, full frame cameras tend to produce images that are better in some ways. They usually produce better images in low light. The images are larger and because of that allow more cropping. Generally, they produce images that can be enlarged to larger sizes.
However, images from APS-C cameras can be very high quality.
I would say that the difference in cost is greater than the difference in image quality.
If size/weight of the camera and lenses is your driver, APS-C cameras are smaller and lighter than full frame cameras.
What do I use?
Honestly, I prefer my full frame camera. However, when I pack up my full frame photo equipment, it is a larger and much heavier set of gear to carry.
Therefore, on some occasions, I take my APS-C camera equipment.
Here are some of the reasons:
When I anticipate lots of harsh weather (rain, snow, wind, sand) my full frame camera is weather sealed. My APS-C camera is not.
If I anticipate lots of walking or lots of equipment moving and pretty good weather, my APS-C camera and lenses are much lighter.
For night photography and for photography in dark places or low light (sunset, sunrise, winter or overcast skies) my full frame camera works better in those conditions.
For fast moving wildlife or action photography my full frame camera works better.
Thinking of purchasing a camera?
Most full frame cameras are very expensive. I use a Sony SLT-a99. This camera was first produced about 6 years ago. Sony has issued a new version (the SLT-a99 Mark II). The newer one is more than 3 thousand dollars. However, you can still find lightly used copies of the original a99 for a $1000. to 1100. dollars.
I bought mine used. It has been a great camera.
Canon has one full frame camera that is older, but still available - the 6D. They also recently issued a newer version of it. But you can find used copies of the original 6D for much lower cost. I have not used this Canon model, but I think you would find it to be a good camera.
There are certain APS-C cameras that stand above others and produce very good images. Here are some of them:
1) The Sony a6000. Nice small, light camera with excellent focusing and great image quality. There are a few reasonabely priced lenses. This camera is affordable and very versitile. (I use one).
2) The Nikon D7100, or the D7200. Good, slightly heavier camera with excellent image quality. Nikon offers lots of lenses that work on this camera.
3) Canon SLT1 or SLT2 cameras are small, high quality cameras. Lots of lenses are available. Reasonabely priced!
4) Canon 70D or 80D. If you want to spend more for a very versitile camera that produces high quality images, these two cameras are very good. I used a 70D for three years and found it to be a great camera. A used 70D would be a better buy than the 80D. The newer 80D costs more, but is a bit faster, with outstanding focusing prowess.
5) Fujifilm X-T10 or X-T20. These fuji cameras are light, but sturdy and produce excellent quality images. There are two or three affordable lenses available. Additional lenses are expensive. If you are willing to spend more and are looking at the Canon 80D, you should check out the Fuji X-T20.