The first type is the fragile compression trigger or "breakaway" trigger. The most basic example is a pit trap covered with a weakened platform or lid, such as balsa wood or glass, which won't support any additional weight (this is distinct from a trap door, which will support weight, but is held in place by a catch which can be opened by a trigger.) Breakaway triggers are usually pretty obvious visually, although sometimes, the floor and the platform are covered with sand, mud, pebbles, or some other substance to hide the exact location; in this case, there is an obvious visual clue that a trigger is present, but another method (such as tapping to locate hollow spots.) Variants involve something that breaks on contact that is not necessarily a platform, such as a weak support column that will break if hit by a door that's been kicked open, causing the ceiling to collapse.
The second type is an indirect compression trigger. A simple example is a loose flagstone placed on top of one or more fragile containers (glass, thin bladder, pottery, or a gourd.) Stepping on the flagstone crushes the containers, releasing their contents, which could be poisonous gas, or chemicals that catch fire when mixed with each other or when exposed to air. For the door variant, a glass flask or other fragile container could be attached to the other side of the door, so that if it slams against something, the contents are released.
Instead of a solid surface, it's possible to use sand, mud, snow, or another substance which compresses under weight. This can be a variant of the breakaway trigger, or the substance can cover a fragile container as above. This is essentially a primitive equivalent to a landmine. Inhabitants of a lair may place multiple "mines" in a room, leaving a path only they would know so that they could pass safely. If multiple adventurers cross such a room one by one using the exact same path, the first person would definitely trigger a "mine", but the second person would have a slim chance to miss any remaining mines; each time a mine is triggered along the same path, the chance improves a little bit more for anyone taking the same path, as the triggers are slowly used up. If the contents of the container leave a "stain" in the sand, mud or snow, or if victims bleed when affected, this could leave a visual clue to the danger.
Not all compression triggers involve breakage. Consider a bellows or a sturdy bladder with a valve; stepping on a flagstone compresses the bellows or bladder, forcing air or some other fluid out, perhaps through a pipe. One simple use for this is to have some kind of fine, powdery poison or dust in the tube, blowing it up into the air around the person stepping on the trap. This could cause coughing, blindness, suffocation, or some kind of toxic reaction. Similarly, a needle or dart could be loaded in the tube to be fired when the bellows is compressed. The more common use for this variant is to use the air stream to trigger something else; this can be one form of pressure trigger, for example. The benefit of a bellows trigger is that it's repeatable, as long as it's not linked to another trigger that has to be manually reset or reloaded. You could design a stack of needles that drop into a tube one at a time, with the tube connected to a bellows trigger, to get a repeating needle trap.
There's not much that can be done to disable a compression trigger. Usually, you would just avoid it, or attempt to trigger it from a distance (rolling barrels or boulders over sandy floors, for example.) Compression triggers linked to doors that are opened too far or too violently are hard to detect and thus hard to avoid, but if adventurers can peer through a crack in the door or the keyhole, or otherwise find some way to see what's on the other side of the door, they may have enough warning to try opening the door carefully. For a bellows trigger, adventurers can locate the tube to cut it or otherwise block it, which would actually disarm the trigger.