What are those large clocks on battleships?

These large 'clocks' are commonly called range clocks, though the British called them concentration dials. Developed towards the end of WWI, they were part of a system to concentrate the fire of several ships onto the same target.

In the days before WWII, the most accepted formation for battleships was the line of battle, where ships would line up nose to tail and fire broadsides at the enemy. The British found that due to the poor visibility in the North Sea, further hampered by funnel and gunfire smoke, it was fairly rare for all the ships in the line of battle to actually see the enemy. So large dials resembling clock faces were added to the front of the fore mast, and to the rear of the main mast, trainable to 30 degrees off center. These clock faces were solid white disks (though a couple of ships had open centers) with black numbers from 0-9. Similar to a clock, there was a short hand and a long hand; the short hand was blue with a diamond at the end, and the long hand was red with a circle at the end.

Once a ship found the range to the target, it would display the range on the dials, with one hand indicating thousands of yards + 10,000, and the other hundreds of yards. So if both hands were on the 1, the range to target was 11,100 yards. Hash marks, called bearing indicators, were painted on the main gun turrets, giving observers the angle to target. With the range and bearing data from the ship in front or behind them to feed into their range keeping tables (fire control computers to us Yanks), and the proper adjustments made for the distance between the ships, their location in formation, etc, the gunnery officers on a battleship could target their guns on an enemy they could not see with a fair degree of accuracy. They could then pass the information on to the next ship in line, allowing the entire battle line to concentrate fire on an unseen enemy.

These range clocks were fitted to British, American, Brazilian, and French battleships and cruisers. However, as first radio communication, and then radio-equipped spotting aircraft allowed for the instant and accurate transfer of targeting information, these clocks were phased out. By the dawn of WWII, only a few ships still had them, mostly the older American dreadnoughts. As each went into the yard for a refit, the clocks were removed.