Cryptography
Encryption that uses a pair of keys - one shared openly, one kept private - so strangers can protect data for you without a shared secret.
Formal
A family of methods in which each party has a mathematically linked pair of keys; data locked with the public key can only be unlocked with the private key, and anything signed with the private key can be checked by anyone holding the public key.
In plain English
Like a mailbox with a slot - anyone can drop a letter in, but only the owner has the key to take letters out.
In practice
A social worker in a municipality sends a case file by encrypted email to a hospital. Her mail program locks it with the hospital's public key, so only the hospital's private key can open it.
Why it matters
It solves the problem of sharing a secret safely with strangers, and makes digital signatures possible, which is what lets people trust who is on the other end.