Public Key and private key

What differentiates PKI (Public Key Infrastructure) from most "shared key" cryptography patterns is the use of TWO keys. The keys, one public and one private, are mathematically linked such that something encrypted with one can only be decrypted by the other (and vice versa). The private key is private because it is meant to be kept a secret. The public key however is public because it is meant to be shared with others that want to communicate with you cryptographically.

A public key is also often referred to as a "certificate" because it contains additional information about the entity and is "certified" by an issuing authority. Here's how that roughly works:

1. An entity will generate a public/private key pair. At this point they're just cryptographic blobs of data.
2. The entity adds information to the public key in a format called X.509 and then repackages the new data into a "certificate signing request" or CSR.
3. The CSR is passed to a certifying Authority, which adds information to the X.509, and then "signs" it - generates a hash of the data, encrypts the hash with its private key, and the incorporates information about itself into the new certificate so that its validity can be tracked back to the authority. The resulting certificate is the combination of public key and X.509 data about the entity and its issuing authority.

SSL is simply a protocol that uses public and private keys to perform cryptographic communications between two entities. Message encryption is performed when a sender encrypts a piece of data with the recipient's public certificate (so that only the recipient can decrypt with the private key). Digital signature is performed when a sender encrypts something with its private key so that a recipient can verify the authenticity of the data (because no other entity would have the same private key). There's of course MUCH more to it than that, so probably better to send you to a more complete reference:

http://en.wikipedia.org/wiki/Secure_Socket_Layer

Now, in BIG-IP terms, the client SSL profile is the server side of the SSL "handshake" with the client, and where you would embed the server certificate and private key. The server SSL profile is the client side of the handshake with the back end server (should you want to re-encrypt).