OpenVZ gives you a container as they call it, which looks like its very own Linux OS. You can login to it, install packages with the package manager, etc. All processes you run, e.g. Apache/PHP, plus MySQL , run in this VM. PHP and MySQL would communicate over a Unix local socket, same as on non-virtualized Linux OS.

Docker appears to run each process, in isolation. So you would have 2 isolated processes (Apache/PHP, plus MySQL) which can only talk to each other over a local network pipe (which should be quite fast with minimal overhead).

(this is my understanding).

The underlying technology(LXC) is quite similar to OpenVZ, the main difference being that LXC comes standard with the Linux kernel while OpenVZ comes as a module. They are both lightweight virtualization technologies that create separate process and memory spaces inside the same kernel. I think OpenVZ is also much older then LXC.