Good question! I work on the ZeroVM team and hope I can help clear things up!
Is ZeroVM providing a container similar to Docker?
No, not really. Docker uses LXC and other kernel features to provide a sandboxed environment. ZeroVM runs entirely in userspace and sandboxes a single application.
Briefly, ZeroVM works by first validating the application to be run and then simply executing it. When executed, the application runs with basically no extra overhead — there's no virtual machine behind it.
In order to validate an application, it must be cross-compiled into a special form of x86 machine code. This special form has the nice property that it can be statically verified to be "safe". Among other things, "safe" means that the code wont jump to addresses outside a certain memory segment provided by ZeroVM. The program can also not call the normal system calls, it can only call a very narrow syscall interface provided by ZeroVM. This validation is taken from Google's Native Client project.
How would I use ZeroVM to run Wordpress? What benefits would doing so provide?
You could in principle cross-compile the C program that is the PHP interpreter. ZeroVM could then start the PHP interpreter in the sandbox and feed it the PHP files that make up WordPress. ZeroVM provides a read-only in-memory filesystem and this could be used to read a SQLite database from. That would give you a read-only WordPress site — not terribly exciting :-)
However, as I see it, this is not the main use-case for ZerovM. Existing database-heavy applications need to be rewritten for use with ZeroVM. ZeroVM is more meant for massively scalable systems where you want to process many data items in parallel.
Imagine you have 1,000,000 email that you need to search through. The mails are stored in a block storage such as OpenStack Swift or Amazon S3. This means that the physical files are stored on some number of storage servers. Traditionally, you would need to pull all the mail down to some number of compute nodes in order to search them. With ZeroVM and its integration with Swift, you can send the code to the data. This is possible because code is small (some megabytes) compared to a bulky virtual machine image, and because it's safe to execute untrusted code in the ZeroVM sandbox.
So ZeroVM is meant for highly-scalable architectures where each request operates on different pieces of data.
For a WordPress site, this could mean that one should store each blog post in a separate piece of data and have a dedicated ZeroVM instance responsible for each post. Currently, the filesystem is read-only, but there are plans to make it read-write and the ZeroVM instance responsible for a given blog post could then handle things like comments. You would need a load-balancer in front that is capable of routing the traffic appropriately. The result is a very different architecture than the current WordPress, but a more scalable one. Actually implementing this is currently left as an exercise for the reader.