The Empire: If I had to describe the Empire in one word, it would be "balanced." The Empire is the jack of all trades, master of none. Most factions have one or more obvious deficiencies to compensate for their strengths, but the Empire does everything reasonably well. They have good quality infantry, solid cavalry, good war machines, and good magic support, but they don't excel particularly well in any of them. Their strength comes from their versatility.
Bretonnia: Knights. Knights everywhere. Bretonnians are all about getting on a horse and mowing everything in front of them down with lance and sword. While the army book is relatively outdated and the game generally skews more towards infantry than cavalry, the basic idea with Bretonnia is to use cheap, poorly trained, poorly equipped peasants to hold your enemy, and then smash into them with your cavalry and cause mayhem that way. To that end, their infantry is very low quality, and they don't have access to gunpowder weaponry, but their knights will mow down whatever's unfortunate enough to stand against them. Think France in Medieval 2 for a comparison.
High Elves: Tied with Skaven for the fastest foot infantry in the game. Generally very lightly armored and only as tough as a regular human (and the humans have access to better armor), the elves will fold like tissue paper, but oh God can they dish out a horrendous amount of damage. Their core infantry, while not as good as some of the later factions, is very high-quality, and their elite infantry is some of the best in the game, albeit it tends to suffer a bit from overspecialization, which is a running theme with elves. They're also tied with the Lizardmen for the best magic faction in the game, and they specialize in denying their enemies access to magic. All elves get access to dragons, but the High Elves have the most variety with three types of dragons to choose from.
Dark Elves: The High Elves' angry cousins, they really need to be played aggressively to work. Where High Elves can be played surprisingly well defensively, with the Dark Elves you really need to get in your opponent's face as quickly as possible, or blast him with magic from afar. Like the High Elves, the Dark Elves have a broad range of magic at their disposal, and their special Lore (a school of magic only they get access to) is all about blasting the crap out of anything that looks at you wrong. While they have a more limited access to draconic mounts, Dark Elves instead get to field monstrous creatures like hydras. They suffer from the same general weaknesses as the High Elves, being very fragile and very expensive, but they will hurt whatever ends up being the target of their wrath.
Wood Elves: Rounding out the trio of things with pointy ears, the Wood Elves are all about speed, maneuverability, and shooting. Almost nothing in the Wood Elf book is well-armored, and even their elite infantry will rarely march around in anything but light armor, but the usual elf-caveat of "they will wreck your face if you get near them" applies. The Wood Elves have access to better archers and fast cavalry than their cousins, though, and whenever they're in a forest their combat ability dramatically increases. They're the bane of factions with low Toughness and high armor (such as Bretonnians) because their arrows and spears are armor-piercing. They have access to the same magic lores as their elven cousins, and are a respectable magic faction. They also have dryads and ents I MEAN TREEMEN to help them deal with their toughness issues.
Warriors of Chaos: Plate-armored Vikings devoted to deities of destruction, perversion, plague and scheming. What more could you want? The Warriors of Chaos have no shooting outside of siege engines, but they don't really need it, because they will make mincemeat out of just about everything they fight. Though expensive, the core units of the Warriors of Chaos can and will go toe-to-toe with most factions' elite infantry, and they only get tougher and more versatile if you dedicate them to a particular Chaos God. They're a reasonable magic faction, though they only have access to the lores of the Dark Gods and are thus more limited in their versatility. They have access to both monstrous infantry and high-quality cavalry on top of their infantry, but as mentioned before, they don't have access to a lot of shooting.
Daemons of Chaos: I'll be honest and say I don't know much about how this army plays. Nobody in my area plays much with it, and thus I haven't gotten a chance to experience it first-hand, but the general idea is a mix of the fragile, high-quality troops you see with Elves and unholy monstrous creatures that can strangle dragons with their bare hands and laugh about it. Very good magic faction if you go for Tzeentch.
Beastmen: Rounding out our Chaos trifecta is the Beastmen and...eh. They're probably not going to be added as a faction due to their nature, so I'm keeping this one reasonably brief: Very strong infantry, kinda crap morale, inspires fear in your enemies (literally) and has good general choices. Weak magic choices holds them back a lot, too.
Dwarfs: Dwarfs are one of the slower armies in Warhammer Fantasy, but they make up for it with their unholy toughness and staying power. You find a place you like, you plant your warmachines, which are superior to just about every other faction, and then you shoot your opponent to kingdom come until they come close enough for you to charge in and waste them. What they lack in speed and mobility they make up for with war machines, runic armaments, and sheer dwarven stubbornness. They are also completely without access to magic, compensating for this with access to magic runes that they can inscribe on weapons and war machines. Completely devoid of cavalry, though they do have gyrocopters to compensate.
Lizardmen: Depending on whether you're talking to an Elf player or not, the magic faction in Warhammer Fantasy. They also have access to excellent infantry that absolutely, positively, refuses to run away, backed up by good light infantry and skirmishers. Oh, and dinosaurs. Did I mention the dinosaurs? They combine the stubbornness of dwarfs with elf magic and infantry that will almost never run away. Even if you eventually lose a combat, your opponent's unit is going to be so decimated it doesn't even matter.
Ogres: Almost all monstrous infantry, ludicrously tough, ludicrously strong. Even their infantry charges like it's cavalry. Very expensive per infantry unit, but will absolutely wreck face on a charge.
Skaven: As fast and as quick as elves, absolutely crap morale. Skaven are built for maneuverability and for nasty surprises. They have surprisingly effective war machines, good gunpowder units, good magic (if limited in the lores they can select), and crap-but-numerous infantry. Like an army of redshirts equipped with arquebuses.
Tomb Kings: One of two undead factions, the Tomb Kings have lots of light infantry, good shooting, a few nasty ambushing tricks and the ability to regain lost infantry via necromancy. In order for Tomb Kings to work you really need to keep your special characters alive, so you really need to keep them safe at all costs, though odds are good that if they die, they'll take the opponent's character with them.
Vampire Counts: The other undead faction, combining cheap unbreakable infantry with flying monsters, special characters who might as well all be named 'meatgrinder' and the ability to create characters capable of both magic and bashing skulls in. Like the Tomb Kings and Lizardmen, they excel in wars of attrition because they can bring their units back to life. However, like the Tomb Kings, if your opponent gets your characters early on, you're borked, so don't throw them into fights they can't possibly win.
And last, but by no means least, Orcs and Goblins: Like Warriors of Chaos they will wreck face in close combat. Unlike Warriors of Chaos, they have access to (mostly terrible) shooting and are really crap at magic. If you want a big blobby horde of close combat beasts rather than the smaller warbands of Warriors of Chaos, this army's for you. Also probably the army whose tactics can change the most depending on which special character you bring.