The turbo downpipe is a vital part of the exhaust on any car. It's of particular importance on a turbocharged engine as it connects directly behind the turbo and so has a big impact on how the turbo performs.
Bigger bore pipework and a less restrictive cat section/de-cat pipe will reduce back pressure directly behind the turbocharger which is a good thing. A turbo with less restriction behind it will spool faster which means less delay between you putting your foot down and the turbo boosting properly.
Better flowing pipework also means you get more exhaust gas out of the engine which in turn means you can get more air through the engine and make more power.
The downpipe section typically includes at least one of the catalytic converters on a car. While a catalytic converter is essential for cleaning the exhaust gases coming out of your engine, replacing this restrictive section with a faster-flowing sports cat or a totally open de-cat pipe will significantly help too.
Normally replacing the downpipe requires an ECU remap as the original map won't be designed to take advantage of the extra flow-rate on offer.