The most well-known PKC algorithm is

RSA; the basic idea is moderately simple and you can read about it in the linked Wikipedia article, but you need to understand a bit of number theory, starting with modular arithmetic (if you don't, that should be your first step). There's also a numeric example.

If the secret prime numbers are p and q and their public product is n=pq, then to encrypt a message you take its representation as an integer and raise it to some power, modulo n. Anyone can do that, but most can't take a power and figure out what the base was. The recipient who knows p and q can find the totient phi(n) = (p-1)(q-1) and with some number theory magic use it to invert the power operation.

Cryptology is a word not used very often and usually only by crypto pros.

And for those who use it, it's not synonymous with cryptography; rather, they use cryptography to refer to developing and using cryptographic techniques, cryptanalysis to breaking them, and cryptology to both.

(is cryptology a word?)

Edit: a little offtopic... does PKC prove, or at least rely on, P!=NP?

If P=NP then there's a polynomial-time algorithm to break PKC. Whether this has practical relevance is not clear; if the best polynomial has order 20, then it's still impossible.