Websockets are an evolution of traditional client/server technology. They create a long-held single TCP socket connection between the client and server, allowing for bi-directional fully duplex messages to be instantly distributed with little overhead, leading to a very low latency connection.
Both the Websocket API as well as native WebSocket support in browsers such as Google Chrome or Firefox means that this technology is widely supported. There are also Websocket library implementations in a variety of other languages, such as Ruby and Java.
Originally, the internet wasn't designed to be dynamic. It was conceived as a collection of HTML pages linking to one another to form a web of information. Overtime, the number of static resources increased and richer items, such as images or video, became essential to the web. Server technologies advanced, allowing for dynamic server pages - pages whose content was generated based on a query.
Then entered Websockets
Websockets represented a new standard for bi-directional real-time communication between servers and clients. The standards first approach means developers can create functionality that is truly cross platform. Since web sockets represent a single TCP socket connection, connection limitations are no longer a problem. Cross domain communication was built in with the connection handshake.
WebSockets don;'t make AJAX obsolete by a long shot, but they do supersede Comet (HTTP Long-polling/HTTP Streaming) as the solution of choice for true real-time functionality. AJAX is still best when making short web service calls, and with CORS support more widespread, it's becoming even more useful. However, if you need real-time bidirectional data streaming, the low bandwidth and latency makes WebSockets the obvious better choice.