I'll start with my personal favorite:
Extremely well-disciplined and reliant on basic troops used en masse. Typical HE players will use anywhere from 2-3 blocks of 15-strong Spearelves or Sea Guard (which are spearelves with shortbows) for every 1000 points in a game. HE are also reliant on their specialist troops, able to field more than any other army (by a small margin). White Lions, Phoenix Guard and Swordmasters are all popular choices, along with Dragon Princes (some of the best knights in the game, it must be said) and either of their chariot types.
HE shooting is, surprisingly, lackluster. You'll find more bang-for-your-buck in Brett Peasant Bowmen, because they're cheaper and just as effective. Usually, HE players will only field archers to fill a Troops slot so they can get their Special units on the table.
With the new army book, all-cavalry HE armies are no longer possible. Silver Helms and Dragon Princes compete for Special slots, and few HE players (if any) waste a Special slot on Silvers when the Princes are a few points more per model and MUCH more effective. A nice cavalry configuration usually involves Tyrion with 9 Dragon Princes sweeping across the board from one flank to another.
HE Magic can be very powerful, but you have to gear your army for it. In 2250pts, this usually means an Archmage with the BoH and a couple mage heroes to hold scrolls.
All in all, High Elves can be relied upon to pass their Ld tests most of the time, and when used right, can hit very hard. They aren't a very forgiving army at all due to their low toughness and crappy armor (for the most part) and can be frustrating to play with for beginning players.
Like the High Elves, Dwarves are very disciplined and will pass their Ld tests more often than not. Unlike the High Elves, they are very durable and their armor is excellent. Also, they have god-like firepower; Dwarf gunlines are universally feared and loathed, right above Empire gunlines. The one thing Dwarves lack is mobility; your opponent is going to get the charge much more often than you do. This really doesn't matter to the Dwarves, however, because of their high Toughness and great armor they don't really need to get the charge.
Bascially, Dwarves will go last in combat but hit like a flaming bus full of kilted Scotsmen when they attack. Don't let people tell you there's no strategy needed to play Dwarves, however. You do need to set up charges by using more expendable units to bog an enemy unit down so you can get the flank charge with some Longbeards...or Ironbeards, I forgot which.
One thing the two armies have in common is that they really don't need to move much. High Elves will always strike first no matter who gets the charge because of an annoying army-wide rule that we pay for on units that don't even need it; Dwarves are going to hit last anyway, so it doesn't matter much to them.