I’ve been asked this questions many times, and it’s easiest to explain it using a metaphor.
If you imagine your spine to be a cable with lots of little wires, then complete paraplegia would be this:
All the wires are broken, and no signals can be sent from your brain to the part of your body below the injury or vice versa. This means that would lose all feelings and would be completely paralysed in that part of your body.
Incomplete paraplegia would be like this:
Some of the wires are broken, others are intact. The result can be very diverse. In extreme cases people become completely paralysed below the injury, but can feel everything. For others the effect is exactly the opposite; they lose all feeling, but can stand, walk and even run.
Most cases are in between, they lose some feeling and control, but not completely. For me personally the effect is that I can feel almost everything, but I have very limited control over my legs. I can stand when I lock my knees and hold on to something, I can swim a little, but I can’t walk or ride a bicycle. Luckily I have not lost my bladder or bowel control, which is a very common problem for many paraplegics. For this I am extremely grateful.