We had a problem with our guest bathroom toilet. When we flushed it, the water drained away very slowly - and we heard a gurgling sound coming from the handbasin. After many attempts using a toilet plunger to clear a suspected blockage, we called in a plumber, who tracked the problem down to a pipe that sits near the toilet. This pipe is white PVC (or similar plastic) about 1½ in diameter, and about 3 feet high. It has a black plastic screw-on cap.
The plumber removed something from it, and said that part was faulty, and would return the next day with a replacement part... this was 3 weeks ago, and he hasn't been back!
Now, my first question is - what is the purpose of said pipe? And secondly, what part needs replacing? I'm losing confidence in the plumber returning, so I need to purchase the part myself.
Thanks!

That pipe with the cap is to vent the toilet. That was put in place of because it or someone did not vent the toilet to the main vent.
Inside it acts like a seal to keep sewer gas out from the main line but when you flush the toilet it opens up to allow the 3" toilet line to suck in air so the sewer line can drain.
When it failed it did not allow air to get into the toilet sewer line and thus creating a vacuum and toilet flushes slow and will eventually clog
You can easily replace this yourself and use Teflon tape to re screw the AAV vent back on
Its an Air Admittance Valve
The big box stores should have one.
Here is how it works

Air Admittance Valves (http://www.toolbase.org/Technology-Inventory/Plumbing/air-admittance-valves)

There are some different styles Here is Lowe's

http://www.lowes.com/lowes/lkn?action=productDetail&productId=29145-143-82678K
Signed 21 Boat

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Many thanks - I'll measure the pipe to get the exact diameter, and get one tomorrow!