Barley malt is typically lumped into one of two categories when used for beer: base and specialty. Base malts make up the bulk of what brewers call the grist—those grains which are used for the mash. They're mainly there to provide the fermentable sugars to be turned into that lovely substance known as ethanol, but they also each carry a distinctive flavor profile. That profile is essential to creating the beer's style, whether it's a Helles or a Dunkel.
Let's start with the basics. The main varieties of base malt you'll encounter are Pilsner, British pale, American pale, and the higher kilned Vienna and Munich malts.* Here's what you need to know about each.
Don't let the name fool you: Pilsner malt can be used in a wide variety of beer styles—not just pilsner. But its delicate flavor and pale color make it prized for the production of clean lagers. The lightest in color of all malt types, you'll see pilsner malt pop up not only in lighter lagers like (yes,) pilsner and Münchner Helles, but also in heftier lightly-colored styles like Belgian strong pale ales.
To taste it, try a maltier Czech-style pilsner—German pilsners carry a bit more perceived hop bitterness, and its the malt we're looking to taste here. Finding an unskunked Pilsner Urquell (look for cans!) would be an excellent way to taste the grainy, honeyed flavor characteristic of pilsner malt.
BRITISH PALE MALT
Typified by the Maris Otter variety, British pale malt, strangely, has nothing to do with cute furry sea creatures. Maris Otter is, however, widely revered as a cornerstone of the English ale tradition. Darker in color than pilsner or American pale malts, this style of malt imparts a touch more flavor, lending itself nicely to the low-ABV, flavorful beers of England.
Wondering what it tastes like? You can pick out the signature biscuity, nutty flavor of the malt in beers like Timothy Taylor's Landlord and Fuller's London Pride (which also contains a touch of caramelly crystal malt).
AMERICAN PALE MALT (2-ROW)
American 2-row pale malt is primarily loved for its high diastatic power—that means it's excellent at converting starch to that sweet fermentable sugar that yeast loves. That said, it is rarely used on its own in commercially available beer (and if it is, its flavor will likely be masked by a mess of hops). This makes it a bit difficult to isolate for tasting purposes, but find a simple American Blonde Ale, and you may be able to get a good taste of American 2-row pale malt, which falls somewhere between pilsner malt and British Pale Malt. Think crackers, grain, and honey.
AMERICAN PALE MALT (6-ROW)
Even tougher to isolate and taste is American 6-row pale malt. This stuff is mainly used because it's even better at converting starch to sugars than 2-row pale malt is, which becomes extremely useful when brewing with non-barley adjuncts like corn and oats. It turns out pale 6-row is less flavorful than any of the base malt mentioned above, and getting a clean taste of it would be difficult in any commercial beer. Try making a beer with it yourself, or at least go taste the grains at your local homebrew shop!
Vienna and Munich malts are kilned at higher temperatures than the other base malts. This gives them a good deal more flavor than the others, and they are most commonly used in small percentages of a beer's grain bill to add complexity. In some beer styles, though, they can form the entirety of the grist.
The lighter-kilned of the two, Vienna, makes up most of the malt used in the appropriately-named but dying Vienna lager style. Negra Modelo is the most widely available commercial example, but the malt character is tainted by the inclusion of a corn adjunct. Instead, look for Vienna's rich biscuity, toasted flavor in the more common Marzen/Oktoberfest beers.
If kilned a little longer at high temperatures, germinated barley will eventually become Munich malt—the key to creating darker bocks, Munich dunkels and most darker Marzen/Oktoberfests. Try Ayinger's Altbairisch Dunkel or Weihenstephan's Korbinian Doppelbock. Both will give you a good idea of the rich flavors that Munich malt can provide. You'll notice a similar, but more intense breadiness than Vienna, with a bit more sweetness and body.
The level of kilning and thus color and flavor produced will vary from malthouse to malthouse, plus, there are "light" and "dark" varieties of Munich malts available to further confuse things.
*Within these styles, malt flavors will vary slightly depending on where the grain was grown and how it was processed. If you're a homebrewer and you want to pick a base malt for your beer, stick to barley grown in the appropriate region for the style of beer you're brewing. The differences in flavor between, say, Belgian and German pilsner malts are subtle, but they'll help you create a stylistically accurate brew.