Chairs and the floor both stop you falling. So can ropes. You can be supported from underneath—you squeeze whatever is supporting you, or from above—you stretch what’s supporting you.
Expect a tension force exerted on something when what’s around it is stretched by the thing. Re-imagine the thing extracted from its environment by choosing to replace the stretched environment by a tension force exerted on the thing.
The re-imagining is simpler, and useful if you choose the tension force to have the same effect as the stretched surroundings that it replaced.
The more the environment is stretched, the greater tension force.
Expect a compression force exerted on something when a solid in contact is squeezed by the thing. Re-imagine the thing extracted from its environment by choosing to replace the squeezed environment by a compression force exerted on the thing.
The re-imagining is simpler, and useful if you choose the compression force to have the same effect as the squeezed surroundings that it replaced.
The more the environment is squeezed, the greater tension force.
You’re most likely to meet compression and tension forces where the surroundings are solids: chairs, tables, ropes. Replacing what’s happening in these surrounding solids with a force means you no longer have to worry about what’s going on in them. But if you decided you wanted to know, here it is.