Judging is a very important role at speedcubing competitions, as the judge you are responsible for ensuring that the solve is legitimate, that there was no cheating happening and all WCA regulations were followed.
The WCA Regulations state that each competitor must be available for judging unless the delegate agrees otherwise, meaning it is important that you know how to do it effectively if you want to compete at speedcubing competitions.
When the runner hands you the cube cover with the cube and scorecard inside, first make sure that the cube is scrambled as sometimes solved cubes can be sent out by mistake, then remove the scorecard and place the cover face down just infront of the timer. for puzzles such as Square-1 and clock, it is important to be especially careful to ensure no moves are made to the puzzle while moving the cover, in the UK we now have special covers to prevent this happening but it is still worth bearing in mind.
Once the competitor says they are ready, lift the cover and start the stopwatch, they have 15 seconds to inspect the puzzle, you should say '8 seconds' and '12 seconds' if they have been inspecting for that long. In the very rare case that they go over 15 seconds then the competitor receives a penalty of +2 seconds if they take between 15 and 17 seconds and a DNF (does not finish) if they take over 17 seconds. Make sure that the competitor does not apply any moves during inspection.
Once the solve is started, you are responsible for ensuring that the competitor is not cheating in any way, you should not look at your phone or be solving your own cube, you should be watching the competitor solve his puzzle, why not think about how their solutions differ to yours? Are there things they do that you could implement to improve your own times?
Once the competitor has stopped the timer, check to see that the puzzle is solved, you can then write the time in the respective box, sign in the judge box and get the competitor to sign in the competitor box, you then put the cube back in the cover with the scorecard and hold it high up in the air for a runner to come and collect it.
If the puzzle is one move away from being solved then it is a +2, if it is more than one move away then it is a DNF, the competitor must not touch the puzzle after the timer has stopped and before the competitor has signed their time (ignore brief accidental touches that do not turn the puzzle).
In blindfolded solving there is no inspection time, the competitor lifts the cover themselves after starting the timer, once they have started memorization, you must make sure that they do not make any moves to the puzzle(s), once they have put their blindfold on, you must then put a piece of paper between the cube and the competitors eyes to make sure they cannot see through the blindfold, the same rules apply for writing down times but in most cases you have to still write the time down even if the cube is not solved (e.g. DNF (2:03.54)).
For a more thorough explanation please read the WCA regulations.