I intended to post this answer a long time ago but I lost the link and didn't find it again until today.

What types of pumps did they use?

The type of pump depends on when exactly, but there were three basic designs. The Burr pump, the common/suction pump and the chain pump. And of course the humble bucket.

The oldest and simplest is the Burr Pump. It consists of a wooden pipe with a one way valve in the bottom so water can enter, but not exit that way. A long rod, called the spear, with a cone shaped leather bucket on the end is thrust down into the pipe and down into the accumulated water. The shape lets the water flow past the bucket when it's going down but traps it above it when it's going up. The spear was thrust down by hand and pulled back up by a rope attached near its midpoint.

I couldn't find a decent drawing or animation so you'll have to make do with my shitty MS Paint creation

The foot of the pump had to be mounted securely so it was placed near the keelsom at the foot of a mast. The top was placed on deck directly above it.

We don't know when it was invented but it had fallen out of favour at the start of the 17th century.

The Common pump started appearing on ships towards the end of the 15th century.

This animation explains its working principle far better than I ever could.

Since a suction pump doesn't have any valves at the bottom it can be mounted pretty much were ever it'll fit though it still has to be above the waterline since the you want the water to go outside the ship and not back into the bilges.

Early suction pumps were made almost entirely from wood. Metal components, primarily lead starts appearing in the late 17th and early 18th century. Copper and bronze appears towards the end of the 18th century.

There were, and still are, many many versions of suction pumps from the simple Ye Olde Village Hand Pump to the multi piston double acting monstrocities of the 19th century. Some variations remain in use to this day.

Suction pumps were the first pumps to be used for washing and firefighting on ships.

The pump they use in Master and Commander appears to be a suction pump.

The Chain Pump appears in European sailing ships during the second half of the 16th century, though the basic construction can be faound amongst the Romans and the Ancient Chinese.

Animation

They could move a lot more water than suction pumps but they required more crew to operate and they required more maintenance. For these reasons they only really became popular on larger warships and they appears to have fallen out of use with the rise of steam power.

Where were they mounted?

The top of the pump were generally placed on either the main deck or, on ships that had them, the gun deck. Mounting them on the gundeck had the advantage of protecting it from the elements and enemy round shot and on since suction pumps can't lift water higher than ~10 metres there were no other option on the biggest ships. On the other hand it made getting rid of the water much more complicated. On the main deck you can just dump the water and it'll find its way overboard in short order.

This isn't an option on the gun deck so you have to provide some way of getting it out of the ship.

Who manned the pumps?

Our governor... had caused the whole . company, about 140, besides women, to be equally divided into three parts and, opening the ship in three places appointed each man where to attend; and thereunto every man came duly upon his watch, took the bucket or pump for one hour, and rested another. Then men might be seen to labor, I may well say, for life; and the better sort, even our governor and admiral themselves, not refusing their turn and to spell each the other, to give example to other(s). The common sort . kept their eye waking and their thoughts and hands working with tired bodies and wasted spirits three days and four nights, destitute of outward comfort and desparate of any deliverance, testifying how mutually willing they were yet by labor to keep each other from drowning, albeit each one drowned whilst he labored.

A Voyage to Virginia in 1609 by LB Wright

Wooden ships leaked. It was a fact of life. The only question was how much. Some ships only required a few minutes of pumping a day while others where kept afloat purely by continuous pumping.

Exactly who manned the pumps during the daily routine depended on the type and size of ship. On a warship it would be done exclusively by common sailors and maybe an unlucky midshipman while everyone was expected to pitch in on small trading and fishing vessels.