I need a hero...
One of the main differences between Warcraft 3 and Blizzard's previous RTS titles is the presence and importance of heroes. Yes, heroes were part of the single player experience in Warcraft 2 and Starcraft, but they were primarily tied to the storyline and were not included in the multiplayer game. In Warcraft 3, heroes are intrinsic to the gameplay in both single player and multiplayer. Heroes not only deal more damage and carry more hitpoints than a "normal unit" but they also have a variety of key spells and abilities that can enhance the units around them. As heroes kill enemies, they gain experience, which allows them to learn new spells, improve existing ones, or add to hero attributes. The upgrade path is decided by the player - it's almost as if your armies are led by Diablo characters!
A human camp
Out with the old, in with the new
In Blizzard's previous War3 design, regular units had to be attached to heroes in order for them to move around the map. They have since decided to scrap this idea and allow players the freedom to use normal units without a hero escort. Blizzard is also considering the idea of not attaching supply credits (the means by which players can create more units) to heroes. As Rob Pardo said, they don't want people to be creating heroes and just stuffing them away in a safe corner the same way Zerg players hid Overlords in Starcraft. Despite all the changes that seem to deemphasize the importance of heroes, Blizzard assures as that the design of Warcraft 3 will be such that players who don't create and upgrade heroes will lose.
Sphere of Influence
One of the big reasons that heroes will be so important in Warcraft 3 is the influence they have on the units around them. Like a great point guard in basketball, or a charismatic quarterback in football, the heroes of Warcraft 3 will allow the troops around them to fight better than they normally would. For example, the human Archmage has a ability called Insight that lets nearby spell casters recharge their mana faster. This will be critical if you're going to rely on the human race's spell casting units. Maybe your elven sorceress will be able to conjure up that extra water elemental. Or your mage priest will be able to recharge just in time to cast that critical "dispel magic" spell. Warcraft 3 will be very much a game of spell and counterspell.
Heroes help you adapt
Speaking of magic, Blizzard also mentioned that any race specific spells will be located on the skill trees of the heroes instead of the regular units. Remember the Ghost's Lockdown ability in Starcraft? That spell wasn't very useful against Zerg, because it only paralyzed mechanical units. So if you were Terran and playing against Zerg, that basically cut down on the Ghost's usefulness. In Warcraft 3, all race specific spells will be located on heroes - players can decide which spells a hero will get as he levels up. If you're human and playing against the undead race for instance, you'll obviously want the Paladin hero's exorcism spell, which does great damage to undead units. But if you were playing against the orcs, you wouldn't need to research exorcism on your Paladin(s).
The general spells reside on the non-hero spell casting units so that they're always useful troops to have, no matter what race you're playing.