"Brine" is what is left after the ISS Urine Processor Assembly processes the urine that is input into it, sending product water to the Water Processor Assembly which makes it potable.
(See the schematic in my answer to How close is the ISS to a closed system, in terms of carbon, hydrogen, and oxygen? for an overall ISS water system diagram)
I have seen various numbers quoted for the percent of water recovered, from 70% to 85%. 93% as quoted in the space.com article seems...optimistic, but perhaps things have improved since I left JSC.
As far as composition, the brine is simply super-concentrated urine.
Here's a schematic of the device.
The original design Recycle Filter Tank Assembly (RFTA) tanks which hold the brine, are returned to the ground for disposal.
The RFTA is or will soon be replaced by a redesigned Advanced RFTA (ARFTA).
In order to support long term ISS operations with reduced up
mass capacity, the RFTA is being modified in order to allow
on orbit servicing of the system. Instead of returning concentrated
brine contained in the RFTA to the ground (or disposing as trash), a
new tank is being developed that is serviceable on orbit. The tank
would collect the concentrated brine solution in the same fashion as
the current tank, but would be removable such that it can drained
into an existing disposable storage tank such as the tanks contained
in the Progress module or Autonomous Transfer Vehicle (ATV).