Empire: has a bit of everything, their infantry and ranged units are well varied and will do well as you go into the late game, their cavalry is also very varied but substantially stronger, you will win by the back of your cavalry. They have also got a huge range of artillery that are exceptionally dangerous if left unchecked, for all their worth though the Empire has next to no flying capability. Empire units represent probably the most generic army in the game though I use that term lightly, if you are are looking for something familiar, the Empire is what you want.
Vampire Counts: boast an exceptional list of extremely strong and numerous infantry and cavalry which are more than equal to any other faction's; this is backed up by excellent flying units and monsters. Vampire Count units are also completely immune to fear, they will never run though if their leadership breaks they will begin to slowly disintegrate. Grim and the Grave (dlc) added a great variety of warchariots that magically enhance your army and debilitate your opponents whilst trampling them underfoot. The weakness of Vampire Counts is their lack of any kind of ranged unit outside of magic.
Dwarves alright listen to me you pole proportioned dendorophile, DWARVES not DWARFS (I'm pedantic) are extremely durable. Their lowest tier units will hold a line better than some of the highest tier units in other factions. They have no cavalry but they do have other things like blasting charges (grenades) and flying machines that drop bombs to supplement that. Their ranged units are second to none and their artillery are not only extremely accurate but powerful in a variety of circumstances.
Greenskins: These guys are almost like the Empire in a lot of ways offering the same unit variety and same restriction of no flying units. Differences start in their durability and damage, Empire units are more durable, Orc units do more damage. Their cavalry is far from as strong as the Empire's and their artillery, whilst still effective is not as effective either. Where they make up for that is in how well the various aspects of their army compliment each other to cover weaknesses and focus on their single greatest strength: getting into a good smashin!
Bretonnia: This is the last vanilla faction yet it's also the latest faction to actually be released so it's a bit hard to evaluate. Their cavalry is probably at this point the absolute most powerful in the game by far which is also supported by flying cavalry of even greater power. After their cavalry they enjoy an excellent variety of units like archers, infantry and artillery which are not as powerful as what other factions can field. Bretonnia makes up for their short comings with their faith and their knights and at present could turn out to be an exceedingly powerful faction.
Chaos Warriors: I'd say these guys are a vanilla faction but that's just not true as they were released as day 1 dlc. These guys enjoy exceptionally powerful infantry, exceptionally powerful monsters, cavalry, monsters and lords. Where they fall short is in their ranged potential, whilst the power of the Doom cannon is absolute they have no ranged infantry.
Beastmen: are amazing, campaign wise not my thing but their army is great. They have a little of everything but enjoy exceptional late game infantry and monsters. They're fast, super fast, their faction is almost the fastest in the game and they love it. The biggest selling point for the Beastmen is their minotaurs which are monstrous beasts that will literally plow through your opponents.
Wood Elves: finishing off you have the Wood Elves, their units are quite strong and in most cases will win in straight combat with equivalently costed units. They have exceptionally strong archers with the highest range in the game, monsters that will never buckle and an excellent variety of powerful ranged units. Their infantry is also surprisingly good and can be generally categorised into either berserker units or Eternal Guards with Shields. They lack artillery though and whilst their Wild Guard cavalry are potent they cannot go toe to toe with higher tier cavalry in other factions.