First point: The things you're referring to are indeed usually known as gullies. The trough description threw me a little at first, which I suspect is why Vicky also mentioned gutters (as in rainwater gutters along the bottom edge of the roof).
Anyway, the use of gullies was pretty much the norm in older houses, and although other methods of connecting to the drainage system can be found in newer homes, gullies are still very much in use.
The grating, as you say, is there to catch leaves, twigs, and any other larger items which might find their way down there. Once you get below the grating, the gulley actually incorporates an S- or P-trap, similar to the type on the bottom of your toilet bowl. It serves the same basic purpose -- A water seal to prevent noxious gases and smells from finding their way back up from the sewer.
You say that poking around the blocked up gulley with a stick it seems to be much deeper than the other. I'd hazard a guess that the grating is missing, so you're feeling right down in the trap. Scum from soapy water and other debris can build up in the trap, and -- as Mindy said -- kitchen waste is the worst as it often contains peas, bits of spaghetti and lettuce leaves, and all manner of other garbage.
You'll need to bail out the mucky water as much as possible (a lovely job!) and then try to clear the trap. Sometimes you can even find that it's parts of the broken grating which have dropped down there and started the blockage.
Once it's clear, it's a good idea to pour several kettles or pans full of boiling water straight down the trap, or even throw some soda crystals down there as well and let it sit a while, then flush through with plenty of cold running water.
