Learn Hardcore Russian: Положи матрас на кровать, пожалуйста.

Положи is the imperative (command/request) form of the verb положить (perfective).
It’s the 2nd person singular imperative, used when you’re speaking informally to one person (to a friend, family member, child, etc.): (you) put / place (it).

What’s the difference between положить and класть (or положи and клади)?

They’re a common aspect pair:

класть / клади = imperfective (“to be putting / to put repeatedly / the process”)
положить / положи = perfective (“to put once, achieve the result”)

In a simple one-time instruction like this, Russian often prefers the perfective imperative: положи = “put it (so that it ends up there).”

Why is it Положи, not Ложи? I’ve heard people say ложить.

In standard Russian, the correct verb is положить (perfective) and класть (imperfective).
Forms like ложить / ложи are widely heard in casual speech in some regions, but they’re considered nonstandard and are best avoided in learning and formal contexts.