Why it is called the Heartbleed Bug?

Bug is in the OpenSSL's implementation of the TLS/DTLS (transport layer security protocols) heartbeat extension (RFC6520). When it is exploited it leads to the leak of memory contents from the server to the client and from the client to the server.
This flaw allows an attacker to retrieve private memory of an application that uses the vulnerable OpenSSL libssl library in chunks of up to 64k at a time. Note that an attacker can repeatedly leverage the vulnerability to increase the chances that a leaked chunk contains the intended secrets.

What makes the Heartbleed Bug unique?
Bugs in single software or library come and go and are fixed by new versions. However this bug has left large amount of private keys and other secrets exposed to the Internet. Considering the long exposure, ease of exploitation and attacks leaving no trace this exposure should be taken seriously.

Is this a design flaw in SSL/TLS protocol specification?
No. This is implementation problem, i.e. programming mistake in popular OpenSSL library that provides cryptographic services such as SSL/TLS to the applications and services.

What is being leaked?
Encryption is used to protect secrets that may harm your privacy or security if they leak. In order to coordinate recovery from this bug we have classified the compromised secrets to four categories: 1) primary key material, 2) secondary key material and 3) protected content and 4) collateral.

More information: http://heartbleed.com/