Here's a rough start that's based on my personal opinion, others can elaborate if they wish:
#1 Pale Malt: Can be used as the base of virtually any style of beer. Will make a blonde beer without much malt character. I find Weyerman's Pale Malt to be tasty enough on its own and have made 1-2 SMaSH Ales with it.
#2 Pilsner Malt: Will create a straw-colored beer with a soft hay-like (to me) aroma. Most famously used in German and Czech Lagers. Can be used in ales, but the character doesn't come through as cleanly. Be sure to boil for 80-90min to drive off all the DMS gremlins (only base malt that always requires this). Often used as the base of Belgian Ales, but I'm not sure if that's just because of tradition.
#3 Munich Malt: Solid malt for malty German lagers and any ale you want a nice, clean malt character (Scottish ales, Dunkelweizens, etc). Like everything else on this list, it can be used as 100% of the grain bill, but it won't have enough diastatic power to convert a lot of other adjuncts, as it has fairly low diastatic power. Add a pound or 2 to your next American IPA if you've never brewed the German stuff and want to get a taste of it. Much milder than its often made out to be, and it will ferment out completely (ie, its not inherently "sweet", just "malty").
#4 Vienna Malt: A sorely underused malt (imho), Vienna (like Munich) can be used as the base of blondes, IPAs, ambers etc. Its German in origin, and gives a rich flavor to beer. Especially in lagers, it jumps out as "toasty" to me when used in large quantities. Can be used as 100% of the base malt. Again, throw in 1-2lbs into your next American Ale if you've never tried it before. Classically, its used as the base of Vienna Lagers, but there aren't too many commercial examples of that style now.
#5 Marris Otter: Classic English base malt for all their malty/toasty/bready ales. More character than Pale Malt, and slightly darker usually. Gives a nice extra dimension to British beers.
#6 Wheat Malt: Hard to describe wheat malt except for that it tastes "wheaty". Soft and rich, very slightly acidic. Very rarely used in lagers, but can be used in ales with either neutral yeast character (American Wheats, wheat wine) or the classic "Hefeweizen" character (Hefe's, Dunkelweizen, Weizenbock). Adds in adding body and head retention in any ale. Sometimes adds cloudiness to a beer, but this isn't as straightforward as you might think. Supposedly can be good in lagers, but I've never tried that.
#7 Golden Promise: Classic Scottish malt, a cleaner version of Marris Otter. Famously used in Scottish Ales, like Belhaven. Less toast, more sweetness than Marris Otter. No tradition of using it in lagers but it would probably be great. Some whiskeys are made with this malt, I believe, but it does NOT have any smoke/peat character.
#8 Plain Ol' 2-Row: Basic malt, slightly less color and character than Pale Malt. Can sometimes need 60+min boils to prevent DMS in the final beer. High diastatic power.
Others that I don't have experience with: Mild Malt, Rye Malt, Optic, Halcyon, 6-Row... I'm sure I'm forgetting a few.