Reference frames in relativity? I've recently been introduced to relativistic celestial Reference Systems (RS) such as the Geocentric Celestial Reference System (GCRS) and Barycentric Celestial Reference Systems (BCRS). So, for example we use GCRS to model near-Earth satellites or use BCRS to model probes far into our solar system. I understand the need for such RS i.e. to be in compliance with the theory of relativity - no favoured coordinate systems, limitation of the speed of propagation of information etc.

However, what are some layman definitions of advantages of such systems to really send the message home? 

I notice that when describing $n$-body dynamics in our solar system for example, Barycentric coordinate time is used instead of proper time. This blows my mind!
It depends on your definition of "layman" but Patrick Wallace has a slide presentation on what changed and why with the new reference systems. There is also the USNO Circular 179 which covers all of this and is pretty readable. 

In a lot of ways it boils down to the realization that the wobbly (in space and rotation) Earth was a lousy thing to base co-ordinate systems on. The new systems are what's known as "kinematically non-rotating" which mean they are fixed in space and don't move. This makes it much easier to make more precise theories (which were needed) to describe how things like the orientation of Earth's pole (the precession and nutation you may have heard about) move in the Solar System without having to reference it to something like the equator and ecliptic which are always moving about.