Monks are great units that convert enemy units, and buildings also but if it is researched. Monks cost 100 gold. Monks can be a key to victory if you capture all the relics you will probably win but if you are playing a game with standard victory condition. Capturing relics also give you gold. Monks are not attacked if they are not doing anything.
Monks are weak vs fast and ranged units such as cavalry and archers, and strong vs slow non-ranged units such as infantry. More monks will capture one unit quickly but they all will lose their faith.
Monks usually start healing units that are in range but you must order them for conversion. Monks can't do anything like military like patrol, guard, aggressive stance etc.
Usage for victory
Winning with monks eg. capturing all relics is very easy.
First of all have your scout explore the map for relics.
Secondly make some military units and have them guard the relics.
Thirdly upgrade your civilization from ages quickly and in a very speed build a monastery, monks and then get the relics. The military units that were guarding the relics now, put them on guarding monks. First get the relic that is near the base of enemy because it is more vulnerable to being catched.
Redemption:Monks can convert enemy buildings, siege weapons except fishtraps, walls, flags etc.
But, this tech is only available in some civilization such as: Saracens, Teutons.
Fervor:Monks will move faster.
Illumination:Monks gain their faith 50% faster after they use it.
Sanity:Monks become 50% harder to kill.
Faith:Little bit costy. Your units are harder to be converted, 50% harder.
Atonement:Monks can convert the enemy monks.
Refer to this if you want to consider civilization techs about monks.
Heavy grip = All techs available.
Medium grip = All but 1 or 2 not available.
Low grip = Almost 3 not available.