Hero Rushing is a standard part of Warcraft III. To Hero Rush means to attack the enemy town, especially workers, as early as possible. Players head to the enemy town as soon as the Hero has been trained. Players are typically either Hero Rushing the enemy or defending against a Hero Rush. Sometimes players may not Hero rush or may decide to Creep instead but Hero Rushing is found in most ladder games.

Reasons for Hero Rushing include

Distraction: People who advance quickly up the tech-tree don't have a strong army. They need to distract and "harass" you until they are able to build their high end units. This involves doing regular hit-and-run attacks on your town. Usually if people are Hero Rushing you and they don't bring units with them, they are teching. You know it's time to go hit their base before they get whatever they are trying to build.
Fun: Hero rushing is fun! It's fun to attack players who don't know how to handle a Hero Rush. If the enemy is really bad at the game you can totally hose them and win the whole game. In Warcraft II and StarCraft there was a "build up" time where there wasn't a lot going on. Many players found this phase of the game boring. The answer was Heroes very early so that players have something fun to do at the beginning of the game.

Good Hero Rushers
Demon Hunter (Mana Burn/Immolation), Blademaster (Wind Walk), Warden (Shadow Strike/Fan of Knives), Keeper of the Grove (Entangling Roots), Far Seer (dogs), Shadow Hunter (Serpent Wards), Archmage (Blizzard the workers/Water Elementals), Mountain King (Storm Bolt - good in 2 vs. 2 games), Naga Sea Witch (Forked Lightning/Frost Arrow), Crypt Lord (Beetles/Impale with Ghouls)
OK Hero Rushers
Some of these might be more effective in Team Games with help.

Blood Mage (Flame Strike), Death Knight (Death Coil to Heal + Ghouls), Tauren Chieftain (Stomp/Shockwave), Beastmaster, Pandaren Brewmaster (Drunken Haze), Dread Lord (Sleep/Carrion Swarm with Ghouls), Lich (Frost Nova), Dark Ranger (possible if you go Life Drain first).

Semi-Bad Hero Rushers
Priestess of the Moon (at least go Searing Arrow first and come with an ally) and Pit Lord (gets stuck). They are ok if they are supporting another Hero and are not being hit.

Extremely Bad Hero Rushers
Paladin (laugh whenever you see this)

Anti Hero Rushers
The same Heroes that are good for Hero Rushing are also good at defending against Hero rushes typically. Some of the best anti Hero Rushers are: Demon Hunter (Mana Burn), Mountain King (Storm Bolt), Warden (Shadow Strike), Far Seer (dogs/Chain Lightning).

Defense against Hero Rushes

All: Lead the enemy away until you can build up enough forces to take them on. Hit, then run away from your town. Use your Town's defenses!
Humans: Get an Arcane Tower! Use Militia if they get close but go back to work if they run away.
Orcs: Hop in those Burrows. Build 1-2 Watch Towers
Undead: Run the Acolytes away. Build 1 Nerubian Tower. Use a Death Knight to heal the Acolytes
Night Elves: Use Detonate. Fight by the Moon Wells. Build your town in such a way that the Ancients of War can