Lightweight virtualization", also called "OS-level virtualization", is not new. On Linux it evolved from VServer to OpenVZ, and, more recently, to Linux Containers (LXC). It is not Linux-specific; on FreeBSD it's called "Jails", while on Solaris it’s "Zones". Some of those have been available for a decade and are widely used to provide VPS (Virtual Private Servers), cheaper alternatives to virtual machines or physical servers. But containers have other purposes and are increasingly popular as the core components of public and private Platform-as-a-Service (PAAS), among others.
Just like a virtual machine, a Linux Container can run (almost) anywhere. But containers have many advantages over VMs: they are lightweight and easier to manage. After operating a large-scale PAAS for a few years, dotCloud realized that with those advantages, containers could become the perfect format for software delivery, since that is how dotCloud delivers from their build system to their hosts. To make it happen everywhere, dotCloud open-sourced Docker, the next generation of the containers engine powering its PAAS. Docker has been extremely successful so far, being adopted by many projects in various fields: PAAS, of course, but also continuous integration, testing, and more.