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Please keep adding your strategies to this guide!
This guide was last updated for the 13w03a snapshot, or the Redstone Update. 
Some information may be outdated if the current version of Minecraft is past this.

A compilation of multiple LPer's strategies on beating a challenge map. Most of this tutorial is based on Complete the Monument styled maps, but will apply to any type of challenge map.

Terms

Some general terms used in challenge maps.

Term Definition
Cheaty hole 1-block wide hole that takes the player into the void.
Dungeon A structure in which there are spawners. In challenge maps, they usually contain a wool chest or other valuable materials.
Birdcage/Fleece/Fleecy Box A box surrounding the Wool chest, usually to protect it from creepers, etc. It typically has a glass encasing and a brick floor.
Funbox A 3x3 box of spawners. Usually covered in stone.
Spawner A Monster Spawner.
Victory Monument A monument where you place your Wool(below definition).
Wool A generalized term for what is needed to "beat" the CTM map. This could be records, actual wool, plants, etc.
Wool chest A chest that will contain Wool(above definition). Usually marked with glowstone.

Useful things to know

These are things that would be very useful to have knowledge about when trying to beat a custom map.

Damage

Main article: Health and Damage
Main article: Enchantments

Mobs become easier to kill when you know how many hits they take with your weapons. It's also good to know how much damage a weapon does if you need to know which one of your weapons is the strongest.

Weapon Damage
Name Damage Dealt
4♥♥
5♥♥♥
6♥♥♥
7♥♥♥♥
1♥ - 9♥♥♥♥♥, rarely 10♥♥♥♥♥
Damage Modifiers
Name Damage Modification
Sharpness Damage is increased by 1♥ to 3♥♥.
Smite Damage is increased by 1♥ to 4♥♥ on zombies, skeletons, and zombie pigmen.
Bane of Arthropods Damage is increased by 1♥ to 4♥♥ on spiders and silverfish.
Fire Aspect 3♥♥ of fire damage with Fire Aspect I, 7♥♥♥♥ of fire damage with Fire Aspect II.
Power 1.5x multiplier with Power I, and increases by 0.25x every next level (Power II = 1.75, Power III = 2, and so on)
Flame 4♥♥ of fire damage.

Fighting mobs

A general list of strategies on how to take care of certain situations.

Large groups of mobs

If there are several mobs in a group, it is best to walk backwards, away from the group (kiting). If there is a creeper within the group, let it explode, and it will kill most of the mobs within the group.

fuck you

Two-hit-kill mobs

These include mobs like Zombie Pigman. If you can, build a barricade around yourself, which will prevent them from getting inside. Take them out by bow, even in close range.

Skeletons

Try and use your bow if you don't have at least a 3-hit kill weapon. As of 13w03b, skeletons will attempt to knockback players in close range by firing arrows quickly.

Blazes

Fire Resistance are extremely useful against Blazes. Don't get too close to them if you have to attack them with a sword, since they deal heavy contact damage.

Ghasts

Ghasts are extremely annoying in maps, because not only do they break blocks and set things on fire, but they also fly away from the player, making them hard to kill.

You want to dispatch of the Ghast as soon as possible, otherwise it'll get too annoying. To kill a Ghast, aim for the body, or deflect its fireballs. This is the ghast's hitbox.

Ghasts cannot break Cobblestone or anything with a higher damage resistance, so use that to your advantage. Always build with Cobblestone, if you have some.

Silverfish

Often, a mapmaker will create a passageway with walls of Silverfish Blocks. If you break one of these blocks, it will spawn a Silverfish, which, when attacked, will spawn even more Silverfish from adjacent blocks. A Silverfish will not call for help if it is killed in one shot by a player, or if it died by "natural" causes.

A Silverfish has 4 hearts of health. Here are some ways of killing a Silverfish without it spawning more Silverfish:

Conquering dungeons

Funboxes

These malevolent boxes cause nothing but trouble. Here's some ways to dispatch of them.

Lava

Lava is the best way to disable a funbox. It lights up the area around the funbox, preventing any spawns, and kills any monsters already spawned by the funbox.

To use lava, get within block-placing range of the funbox, and pour lava on it.

If you cannot reach the area of the funbox, build up so that you are a little higher than the funbox, and pour lava down onto the funbox.

Cannons

Using a TNT Cannon is a fun way to dispatch of funboxes. However, using these could blow up a wool chest, not to mention yourself. Be careful with this method.

Tunneling

The easiest way of tunneling toward a funbox is to go from below, as most funboxes are placed out in the open. Make sure to light up your tunnel very well while you tunnel towards it: you don't want your tunnel to be too dark.

If in a dungeon, always break the funbox before moving on. Creepers are known to blow up your lighting around the funbox and start it up again.

Big rooms

Rooms that are generally, very big. They are difficult to light up because you have monsters attacking you from all directions.

Runthrough

Drink a Potion of Swiftness, and charge into the area, placing torches where you can. This is very dangerous, but very effective as well, as you will be covering large areas, and disabling lots of spawners.

Charge and retreat

Charge in, light up the area, and retreat. Effective, but snail-slow, and rather dangerous.

Tunneling

Tunnel around the room, lighting up the walls of the room at regular intervals. This will disable most of the spawners and will make the Charge and Retreat method much easier. (It will be more effective to use Lava to disable the spawners, rather than torches.)

Tunneling to the top of the dungeon and pouring lava from above is a much safer way of doing this.

Landscapes

Even bigger than a big room, they are vast areas with several little dungeons for you to conquer. As well, there are usually areas on the other side of these landscapes, making these areas much more annoying.

Transportation

Using Minecart Tracks is a very effective way of moving around the landscape. You could also make a tunnel under the landscape for safe travel at night, but this takes a lot of time, and is not worth it.

Void rooms

Areas which open up to The Void or Lava. If you fall off of the main platforms, you will lose your inventory most of the time.

Go slow and steady in these areas. Don't take unnecessary risks. Block off any holes to the Void that you can reach, and be very careful of sand or gravel.

Bridging

If there is a gap between two platforms, make a very secure bridge. Place blocks to the left and right of you, in addition to below you.

If there are too many mobs on the platform where you are heading to, it is best to make a Piston bridge, if possible. Place a Piston facing the platform, and repeatedly have it push blocks. Do this three times: one for the floor, and two for the rails on both sides.

Mazes

Bedrock mazes are found occasionally in maps. They're frustrating to solve and hard to get out of. Here's some tips of finding the right path:

Dead end mark-off

Whenever you hit a split road, put down a torch, and then go left, or whichever direction is closest to left. Always go left. It's much easier to find dead ends if you stick to one direction.

If you find a wall, good. Traceback to where you started, and mark off that path as a dead end (either with stone blocking the way, or a torch on the beginning of the path). If you find a split road, do the same thing: mark off the split road with a torch, and go to the left choice.

Fighting in mazes

A common trick for mappers to do is to put spawners within the mazes, making it harder to move around. Use corners to your advantage when fighting in a maze. Don't use lava: it'll gum up the corridor too much.

What to watch out for

Sand and gravel

In general, a sand or gravel trap is a trap involving floating sand or gravel. When you place a block or a torch on it, it will fall, which will either reveal lava, a pit of monsters, or fall onto a Pressure Plate, which will activate some redstone.

Sand traps that expose areas are unavoidable, however, you can avoid the redstone-style sand trap by being wary. If you see a chest, furnace, or something of that sort, make sure you have a way of getting away before breaking it.

TNT

If it's too good to be true, it probably is. Look out for innocent houses built on the landscape: it could be a deadly trap! TNT also makes a great big hole, making it rather difficult to fight. Also some maps contain TNT which acts like it did in beta, it will detonate on hit making it impossible to disarm. Map makers achieve this through external tools.

Pits

Although easy to dodge within corridors, fighting can distract you, and you can walk off the edge of a pit! Cover up, or barricade every pit you see that might be a threat. If possible direct the fighting away from any nearby pits.

Redstone-related Items

This includes Pressure Plates, Dispensers, and Pistons. Redstone circuits can be made to do virtually anything, so if you see an "innocent" Pressure Plate lying about, block it off, or avoid stepping on it.

Movement-affecting blocks

These include Soul Sand, Ice, and Vines. Soul Sand makes you walk slower through an area; Ice makes the ground slippery; Vines can be distracting when fighting against walls. Try to stay away from Soul Sand or Ice, and avoid being cornered by mobs in a vine-filled area.

Preparations

Your inventory

It should contain:

Two Swords
A bow
 -> up to a stack of Arrows
Two Pickaxes
A stack or two of Torches
Buckets (if you have some)
 -> A Water Bucket is good, as well as a Lava Bucket
Armor (very important!)
A stack or two of Blocks
Wood
Some Coal (or Charcoal) for Torches

Regeneration, Speed, or Fire Resistance potions are also nice things to have, if you can afford them.

In case of an emergency

Oh no! You died in an area with a high monster concentration, and you had your best items! What will you do?

Be prepared for this scenario. Always have an emergency chest stocked with pre-made resources, just in case. It should have minimal inventory: a sword, a bow, some torches, and some armor. Note that you are not trying to conquer the dungeon with this, you are simply trying to get your items back.

The base

A main base should have these characteristics:

- It is completely safe from monsters.
- Crafting Tables, Furnaces, and Chests is situated there.

Generally, most people build their bases around the Victory Monument, but some mapmakers have been annoyances within the monuments (e.x. dying zombies, growing vines), so it is best to build it relatively close, but not too close.

Some more things I recommend you have in your main base:

Food farm

In most maps, the best renewable types of food are Carrots and Potatoes. They are very easy to farm, and they feed a decent amount (Carrots heal 2 Food Points, Potatoes heal 3 Food Points.) Here is a tutorial on Automatic Wheat Farms, which can be used to grow carrots or potatoes as well.

If there are cows or mooshrooms in the map, then it is best to get a wheat farm started in addition to the carrots/potatoes. When you kill a cow, they will drop Raw Beef, which, when cooked in a furnace, is a much better source of food than carrots, healing 4 Food Points.

A Melon farm, unless very big, is not recommended, as melons only feed one part, and melons grow rather slowly.

Monster farm

If possible, set up a spawner trap around a funbox. If you can't find a funbox near your base, set up spawning platforms. Make them really high in the air. This is to ensure they spawn.

The best kind of trap is a lava blade trap, which kills monsters and leaves their drops unharmed.

A gravity trap takes more time, but is also more effective, as there is absolutely no light near the spawners, giving full spawn rates.

Tree farm

Wood is undeniably the most important resource in Minecraft. You'll need a lot of it too, in the case of challenge maps.

Cobblestone farm

Main article: Tutorials/Cobblestone Farming

This is only needed in extreme cases, where you have a great lack of Stone to use. The easiest farm to make is

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