As the name implies, bittering hops add bitterness; without them beers tend to be unbalanced*; lacking the bitterness to balance out the residual sweetness of fermented wort.
*there are alternatives, discussed below
The bitterness comes from 'alpha acids', a group of chemicals in hops which taste bitter. These are normally insoluble in water, but heating isomerizes them, making them soluble in the beer. This is why bittering hops are usually boiled for 60+ minutes; less than that and you get poor isomerization/extraction of the alpha acids. This long boil drives off many of the flavour and aroma compounds, leading some brewers to claim that all bittering hops are the same.
As you mention, some brewers (myself included) feel there is a difference between bittering hops. Some leave an unpleasant astringency, grassy flavors, etc. Just how prominent these flavors tend to be dependent on the amounts used, the time in the boil, and other factors. Exactly what makes those differences, or even if they exist, is not well established. One possibility is that some strains of hops may contain more easily extracted tannins and polyphenols, which tend to have a harsh edge to them.
Now for the *. A method you'll likely encounter is 'hop bursting'; using huge amounts of hops in the last 15-20min of the boil. This is a way of packing a lot of hop flavour into a beer, while avoiding excessive bitterness. You still get alpha acids/bitterness, but instead of using a long boil to get efficient extraction you rely on inefficient extraction from a massive amount of hops added late in the boil. People report that this method produces a smoother & more enjoyable bitterness; I'd agree with the former claim, but I always add a little bit of bittering hops to my hop-bursted beers as I find a pure-bursted beer is too mellow for my tastes.
A third thing you may run into is first wort hopping; this is adding hops to the run-off of the mash, before the boil (and then heating/boiling the hops). This, again, produces a different bitterness profile. I really enjoy the slightly different (smoother, but taste-forward) bitterness profile this produces and thus tend to employ it.
I know that's a lot, and I described everything in very basic terms. Long story short - once you're comfortable with the basic brewing process, try some different bittering hops and hopping methods. Keep notes, and find what you like!
Bryan
