The TITANIC used two four cylinder triple expansion reciprocating engines (or "reciprocating engines" for short), and one low pressure turbine engine. OK, I bet that at least 97.85% of you have no idea what that means. So I'll break down each part and explain it seperately to give you a better idea, but to save time let's start right at the beginning with what happenes on the bridge.

STEP #1: The Bridge On the bridge of the TITANIC there were three telegraphs. Telegraphs are devices with phrases on them such as; ALL AHEAD FULL, AHEAD EASY, ALL STOP, FULL ASTERN, etc. Each phrase described the desired speed of the ship. Messages would be sent to the engine and boiler rooms via these telegraphs, and the engineers would carry out the instructions given. So let's say that we wanted the ship to travel at full speed (24knots). We would move the handle on the telegraph until the arrow pointed at ALL AHEAD FULL. ding-a-ling-ding-ding-ding The arrows on the receiving telegraphs down in the engine and boiler rooms would then ding to ALL AHEAD FULL, and someone would ding (incase you haven't figured it out by now, the telegraphs had bells in them that rung when orders were given) the handle of that telegraph to ALL AHEAD FULL, signalling to the bridge that the message has been received and carried out.

STEP #2: The Boiler Room The boiler room would then shovel a certain amount of coal into the furnaces and change the pressure of the boilers. The furnaces heat the water making it into steam, then the steam travels to the engine room.

STEP #3: The Reciprocating Engines The engine room is where those four cylinder triple expansion reciprocating engines come into play. The steam is pressurized some more, and then enters the first cylinder of one of the two reciprocating engines. The first cylinder of four is the one with the highest pressure and is called the high pressure cylinder, simply enough. It is here where the steam is put to work. By passing through the cylinder, the steam must push a piston out of the way. When it does this it loses some of it's pressure, and enters the second intermediate pressure cylinder. The same thing that happened in the high pressure cylinder happens in the intermediate pressure cylinder except with a bit less force. After the intermediate pressure cylinder, the steam enters the first of the two low pressure cylinders. As you may have guessed, the same thing happens here except with even less force. Then the steam enters the second low pressure cylinder, does the same thing.
When the steam pushed the piston out of the way, the piston forced a vertical shaft downwards. This shaft was connected to a giant crank, which was connected to a horizontal shaft, which was connected to the propeller. Since the crank was connected to the horizontal shaft, instead of just being pushed down and through the ship, it turned the horizontal shaft. The vertical shaft would swing back and forth in order to let the crank turn. The horizontal shaft was connected to the propeller, so it turned the propeller as it turned.
The waste steam used by both reciprocating engines would then travel to the low pressure turbine.

STEP #4: The Low Pressure Turbine There was only one turbine on the TITANIC, unlike the two reciprocating engines, and the turbine worked in a much different manner as well. As far as I know, she worked kind of like a paddle wheel. There was a paddle-wheel-type-thing inside of a casing, and the steam would travel through the casing pushing the paddle-wheel-type-thing out of the way and turning it. The turbine was connected to a shaft that was connected to the centre propeller. When the turbine turned, the centre propeller turned with it. Unlike the wing propellers, the centre propeller couldn't turn backwards because the turbine worked only in one direction, unlike the reciprocating engines.

STEP #5: The Condensor After the steam passed through the turbine it was sent to the condensors, which would condense the steam back into water for the boilers.