The professors award and take away points to houses simply by stating the fact. e.g. "10 points for Gryffindor". I don't recall it ever being said that someone wrote that down or anything. Do the books ever explain how all those points are tracked? I'm sure it involves some kind of magic but are the details specified?
The answer is by magic. My understanding is that anyone with the authority to reward or deduct points (teachers, prefects, Head Boy, and Head Girl) upon declaring the points adjustment the system automatically tabulates it.
From the dubious Harry Potter Wiki:
Exactly how points are tabulated and recorded is not explained in detail. It is implied in several books that simply stating the addition or subtraction of points magically adjusts the score accordingly. It's not certain if this occurs in all cases, such as when Dumbledore adjusts the scores at the end of Harry Potter and the Philosopher's Stone. The four giant hourglasses, however, seemingly adjusts their contents accordingly to the house points granted or retracted by the teachers' wordings.
I'm fairly certain there is a point in the book when a professor docked points and the students saw it was immediately reflected in the hourglass for that house, but I don't have the reference.
There are hourglasses with jewels in them to document the points. In Harry Potter and the Order of the Phoenix, Harry is near the hourglasses when Snape tries to take ten points from him and sees that Gryffindor has no points left. McGonagall then walks by and gives Harry, Ron, Hermione, Neville, Ginny, and Luna fifty points each for their actions at the ministry, and a large amount of rubies fall into the bottom of the Gryffindor hourglass, and sapphires fall into the bottom of the Ravenclaw one. Then the ten points Snape wanted to take from Harry cause ten rubies to fly from the bottom of the Gryffindor hourglass to the top.
‘Right then,’ said Professor McGonagall, looking up at the hourglasses on the wall. ‘Well, I think Potter and his friends ought to have fifty points apiece for alerting the world to the return of You-Know-Who! What say you, Professor Snape?’
‘What?’ snapped Snape, though Harry knew he had heard perfectly well. ‘Oh – well – I suppose …’
‘So that’s fifty each for Potter, the two Weasleys, Longbottom and Miss Granger,’ said Professor McGonagall, and a shower of rubies fell down into the bottom bulb of Gryffindor’s hour-glass as she spoke. ‘Oh – and fifty for Miss Lovegood, I suppose,’ she added, and a number of sapphires fell into Ravenclaw’s glass. ‘Now, you wanted to take ten from Mr Potter, I think, Professor Snape – so there we are …’
Harry Potter and the Half-Blood Prince / CHAPTER THIRTY-EIGHT — The Second War Begins