How do I know when to use who or whom? I never understood this.

The words usually introduce a clause, and 'who' is the subject in the clause, while 'whom' is the object.

I don't remember who kissed me. ('Who kissed me' is like little sentence on its own; subject-verb-object.)

I don't remember whom I kissed. (The word order is different from a regular sentence here, but; object-subject-verb.)

I don't remember who I kissed. ('Who' can be used instead of 'whom', and usually is, especially in spoken English.)

'Whom' is formal and old-fashioned. When writing a clause, you can put a preposition in front of 'whom'. "I don't remember to whom I spoke." But if you use 'who' as an object (as most people do), you can't. "I don't remember who I spoke to."

I've always used this little trick (mainly to avoid rembering the rule). Ask yourself how you would answer the question without using the word "was."

If it can be answered by using "he" or "she" the proper word to use is "who."

I don't remember who kissed me. HE kissed me. Not, HIM kissed me.

If it can be answered by using "him" or "her" the proper word is "whom."

I don't remember whom I kissed. I kissed HER. Not, I kissed SHE.

is not a sentence unless it ends with a ?

Unless, cattywampus, "who" is a name, as in "Cattywampus kissed me."

But, even if it is not a name, what is your point?

Costello: "I throw the ball to who."
Abbot: "Exactly."

in traditional english, meaning proper and not colloquial, who is the subject, and whom is the object, both direct and indirect.