Let me explain with an example.
In normal key-pair based PKI, there are private key and public key.
In a certificate-based system, there are private key and certificate. Certificate holds more information than the public key.
Demo (You can generate a certificate and private key): http://www.selfsignedcertificate.com/
You can download open the private key file and certificate file, you see certificate file contains much information as shown below.
You can match your generated certificate (opening by a text editor), and private key (opening by a text editor) from this site: https://www.sslshopper.com/certificate-key-matcher.html
If the certificate matches client's private key, the client is sure, that certificate is given by the client or given by client's trusted agent (CA).
However, there are problems in only private key and certificate-based communication.
Because, anyone can generate their own certificate and private key, so a simple handshake doesn't prove anything about the server other than that the server knows the private key that matches the public key of the certificate. One way to solve this problem is to have the client have a set of one or more certificates it trusts. If the certificate is not in the set, the server is not to be trusted.
There are several downsides to this simple approach. Servers should be able to upgrade to stronger keys over time ("key rotation"), which replaces the public key in the certificate with a new one. Unfortunately, now the client app has to be updated due to what is essentially a server configuration change. This is especially problematic if the server is not under the app developer's control, for example, if it is a third party web service. This approach also has issues if the app has to talk to arbitrary servers such as a web browser or email app.
In order to address these downsides, servers are typically configured with certificates from well-known issuers called Certificate Authorities (CAs). he host-platform (client) generally contains a list of well known CAs that it trusts. Similar to a server, a CA has a certificate and a private key. When issuing a certificate for a server, the CA signs the server certificate using its private key. The client can then verify that the server has a certificate issued by a CA known to the platform.
However, while solving some problems, using CAs introduces another. Because the CA issues certificates for many servers, you still need some way to make sure you are talking to the server you want. To address this, the certificate issued by the CA identifies the server either with a specific name such as gmail.com or a wildcarded set of hosts such as *.google.com.
The following example will make these concepts a little more concrete. In the snippet below from a command line, the openssl tool's s_client command looks at Wikipedia's server certificate information. It specifies port 443 because that is the default for HTTPS. The command sends the output of openssl s_client to openssl x509, which formats information about certificates according to the X.509 standard. Specifically, the command asks for the subject, which contains the server name information, and the issuer, which identifies the CA.
$ openssl s_client -connect wikipedia.org:443 | openssl x509 -noout -subject -issuer
subject= /serialNumber=sOrr2rKpMVP70Z6E9BT5reY008SJEdYv/C=US/O=*.wikipedia.org/OU=GT03314600/OU=See www.rapidssl.com/resources/cps (c)11/OU=Domain Control Validated - RapidSSL(R)/CN=*.wikipedia.org
issuer= /C=US/O=GeoTrust, Inc./CN=RapidSSL CA
You can see that the certificate was issued for servers matching *.wikipedia.org by the RapidSSL CA.
As you can see, because of this additional information sent by CA to Servers, the client can easily know whether it is communicating with its server or not.
Certs used for SSL is heavily based on PKI http://en.wikipedia.org/wiki/Public-key_cryptography
– Zoredache – 2013-07-15T18:39:41.643