Public-key cryptography refers to cryptographic systems that require two different keys, linked together by some one-way mathematical relationship (which depends on the algorithm used, but in any case the private key may never be recovered from the public key).
Typically, the private key is used to decrypt (and/or sign) a message, and the public key is used to encrypt a message (and/or authenticate it – that is, check its signature). Before beginning communications, there must be a key exchange (every involved person send to other people their public key).
The private keys must remain private, otherwise everyone can impersonate the owner of the leaked private key, and everyone can decrypt private messages supposedly sent only to him.
As long as no private key is leaked, the security of the system is ensured: while everyone can check that a message really comes from a specific person (authenticity), no one can fake it. In the same way, while everyone can encrypt a message to some specific person, no one might decrypt it but this person.
Before Alice and Bob can communicate privately and ensure trust, they have to exchange their keys: Alice sends her public key to Bob, and Bob sends his public key to Alice.
Now, Alice wants to send a message to Bob.
She encrypts it using Bob's public key, and adds an authentication tag using her own private key. When Bob receives it, he will check the message's integrity, then he will decrypt it using his own private key.
An adversary has to get hold of Alice or Bob's private keys in order to decrypt this message, or construct a different, valid message.