If the bowl overflowed, it's blocked. We'll check the pan, the trap, and the waste pipe. If the cistern's overflowing, we'll check the ballvalve (is it shutting off?) and the flush valve (is it sealing?). Different faults, different fixes.
We'll use a plunger, auger, or drain rods to clear the blockage. Most toilet blockages are in the pan trap (the U-bend inside the toilet) or just after it. If it's further down the waste pipe, we might need to rod it from outside or use a drain snake.
If the cistern won't stop filling, the ballvalve's usually worn or scaled up. We'll replace it with a new one. Modern ones are quieter and more reliable than the old brass ballcocks. Takes about 20 minutes.
If water's running from the cistern into the bowl constantly, the flush valve seal (the rubber washer at the bottom of the cistern) has perished. We'll replace it. If the whole flush valve is old and crusty, we'll replace the complete unit.
Once it's fixed, we'll flush it a few times to make sure it fills and stops correctly, doesn't overflow, and doesn't leak. We'll check the overflow pipe outside (if there is one) to make sure it's not dripping. Job done.