Yeah, there's lots of threads on this but I'll give you a summary. EDIT: Also, I realized this requires a knowledge of what a wastegate is so I'll start with that.
A wastegate is a valve that re-directs exhaust flow from going through the turbo. This is how your car regulates boost. At maximum PSI, it will re-direct some of the exhaust gas around the turbo so that it doesn't continue to spool. Wastegates then generate a seperate exhaust stream from the main turbo exhaust. Turbos can either be externally gated or internally gated. If the gate is internal, than a bypass flow path is built into the turbo itself (this style is stock on a WRX).
Turbos have two different outputs. The turbo outlet on your WRX is internally gated (meaning inside the turbo) so the backside of the turbo has two wholes for exhaust. The turbo itself has the main opening but there's also a whole where the wastegate dumps out.
The stock WRX downpipe is very crappy, it just has a flat plate connection and then a tube going down. Particularly the wastegate exhaust has trouble flowing straight out of the turbo. This can cause a decent sized power bottleneck.
A Bellmouth design just creates a larger funnel to more gradually and evenly merge the two exhaust streams. Since the WRX is still a 4-cylander anyway, it's usually more than sufficent to milk out the performance gains.
Divorced wastegate means the pipe is actually separate. It has separate exhaust pipes that run from the wastegate and the turbo. This is kinda silly on an internally gated turbo but for bigger turbos that require external gating, it is almost required. The wastegate exhaust pipe and main exhaust pipe are then either run to two separate places (dump pipe) or merged again somewhere further down the exhaust to minimize turbulence from combining streams.
04 STi 16G powered lagless fun
04 WRX: SOLD