Wolfram's 2,3 turing machine was proven to be universal

http://www.wolframscience.com/prizes/tm23/solution_news.html
http://www.wolframscience.com/prizes/tm23/TM23Proof.pdf

A system is "Universal" if it can, given infinite memory and an appropriate program, compute any computable function. A previous system even more simple than this one, Rule 110, was proven to be Universal by one of Wolfram's associates (Wolfram had the idea that it might be, Matthew Cook discovered the proof). However, a Universal Turing machine has some extra requirements with regards to the implementation. So this is the simplest Universal Turing Machine.

If you already know programming, then here's how to think of it:

A Turing machine is a programming language.
A Universal Turing machine is a language that is sufficiently flexible enough to perform any computation (including emulate any other Turing machine - i.e. language)
A Non-Universal Turing machine is a language that is built to do a specific purpose well, but does not have enough flexibility to perform any computation.

For computer programmers, nearly every general-purpose language we deal with on a daily basis is Turing-complete. An example of a non-Turing Complete language might be a configuration file. It has a language, you may even be able to do some basic scripting in it, but unless it is built out of a general-purpose language you cannot perform any computation in it.

What they think it is useful for is to help us to make nanomachines easier. If we can construct a Turing machine with simpler parts, we can have a compiler that can pick up the slack in making the program.