Trap

In IVAN there are only two types of true traps.

If a trap is active, it is hidden from view, unless placed by your character. However, you can detect these hidden traps by Searching and thus can avoid them. All traps can be used to efectively dismember your enemies, if you fel so.

Note that planting any trap inside a city (Attnam or New Attnam) will likely result in an injured resident and subsequently all the population trying to slaughter you.

Beartrap

These are usually found in the Underwater Tunnels levels, but are also found in Gloomy Caves, but not quite as often. The further into the game you are, the stronger the material the traps are made out of. The ones found in the Secret Rooms generally tend to be made of Adamant.

The beartrap's main purpose is to clamp onto your leg, damaging it and prevent you from moving, however if it is made of a strong enough material, it can simply sever your limb. Sometimes when planting a beartrap, you can get caught by it, the chance of this happening increasing while burdened, stressed, exhausted or fainting. Limbs can also get caught whilst trying to escape from the beartrap. Beartraps can easily be used against enemy monsters.

You can deactivate beartraps by kicking them, or picking them up whilst levitating, however, picking them up has a chance of getting a limb caught.

Landmine

Mines are generally found active in Gloomy Caves, and unactive ones can sometime be found in the Tunnels. Mines are deadly to any IVAN character lacking fire resistance, and stepping on one during the early stages of the game usually results in death.
To be safe, a ring or cloak of fire resistance is recommended to reduce the damage caused by the explosion.
A ring of searching can also help, as it will detect the mine and alert you before you step on it.

Currently there is no known way to deactivate mines, but you can destroy them. Praying to Silva, zapping a wand of fireballs, using the Holy Banana or getting a monster to step on it are a few ways to destroy mines whilst keeping safely out of range of the explosion.
Attempting to pick up an active mine (i.e. while levitating above it) is strongly discouraged.

Mines also come in two variants, a landmine and a big mine. The big ones are much more deadly than the standard mines and have a larger blast radius, but do not differ mechanically.

Non-Traps

There are other items which can be considered traps but are technically not because they cannot be actived and are not hidden.

Banana Peel

These can be dropped on the ground and can cause anything that walks over them to slip over and fall down like in all those old cartoons. They are reasonably useful earlier in the game, and can be used to kill all the Banana Growers in New Attnam, and are also very useful for killing those annoying Hedgehogs in the Tunnels.

Broken Bottle

Similar to the banana peel, these have a chance of damaging whatever walks over them. However, unlike the banana peel which can hurt any part of the body, the broken bottle damages the legs first or any other body part that touches the ground.

Backpack Full of Gunpowder

See here for more details.

Not a trap but tool really, they have many uses nonetheless. For one, if you place a few on top of a mine that you planted yourself, they will almost certainly kill anything that stands upon the mine because of the united explosion. This is even better with big mines. Also, sometime when you throw a backpack full of gunpowder at an enemy, they will pick it up, which makes them easy prey for the wielder of a wand of fireballs. If placed on the ground, a nearby explosion will set them off, causing a chain reaction. This is very useful for clearing out hoards of enemies, and is VERY useful in the Big Room floor in Caves.

Mirrored Chest

Hardly a trap at all, there are sometimes chests placed in the dungeons which contain extremely rare and precious loot (think an amulet of life saving, Valpurium flaming sword +3 and a cloak of invisibility in a single chest). You could consider yourself lucky as hell, but these items are nothing but a mirrored copies, placed there to tease you. They will vanish soon, so make out of them as much as you can.

Trapped Door

Arguably could be also categorized as a normal trap, trapped door will detonate if you try to open them. The explosion triggered will be exactly the same as of a landmine, so all information on mines applies on the trapped doors as well.

Also of note is a glitch in calculation of the damage done by an explosion: The damage done is decreased the farther you are from the explosion. However, the distance is calculated orthogonally and thus standing diagonally to any door while opening them will decrease the damage done if they are trapped, as the game will think you are two tiles away from the explosion rather than one.