Learning about cryptography can be discouraging. You get so bombarded by "don't invent your own", "you're doing it wrong", and "even really smart people screw this up" that you wonder why you even bother to try. For me, the answer is because if you don't learn it, someone who knows even less than you will end up implementing it (badly, but with confidence) on your project. So despite being fraught with peril (because I am not an Expert), I'll share a little about the concepts of public key cryptography in a form that has been helpful to me.

"Traditional" or symmetric encryption works like this:

# encrypt
ciphertext = encrypt(message, secret_key)

# decrypt
message = decrypt(ciphertext, secret_key)

Public key cryptography (asymmetric encryption) is so named because there is an additional key beyond the one you keep secret in traditional algorithms. You have a key generator that gives you both keys, like this:

public_key, private_key = keygen()

The private key needs to be kept secret, just like with symmetric encryption. The public one can be shared with the world. Print it on a T-shirt if you like. In fact, anyone that wants to send you an encrypted message needs your public key. It looks like this:

# encrypt
ciphertext = encrypt(message, public_key)

# decrypt
message = decrypt(ciphertext, private_key)

In practice, though, public key cryptography is prohibitively slow, so you'll combine it with a faster symmetric algorithm, so that the process looks like this:

# encrypt
symmetric_key = random()
ciphertext = symmetric_encrypt(message, symmetric_key)
encrypted_symmetric_key = encrypt(symmetric_key, public_key)

# decrypt
decrypted_symmetric_key = decrypt(encrypted_symmetric_key, private_key)
message = symmetric_decrypt(ciphertext, decrypted_symmetric_key)

In this case, when sending the encrypted message, you'd include the encrypted symmetric key along with it.

Aside from encrypting and decrypting messages, public key cryptography can be used to digitally sign a message and verify the signature. The signature for the message is created by encrypting a hash of the message, but using the private key for the encryption instead of the public key. Only the holder of the private key can sign, but anyone can verify the signature with the public key. This looks like:

# sign
signature = encrypt(hash(message), private_key)

# verify
is_valid = decrypt(signature, public_key) == hash(message)

Update: Speaking of being wrong, corrections made to the mixed symmetric/asymmetric example on 24 Jan 2018.