There is no reason to ever use a standard attack as a Barbarian, or any other character. The design philosophy that the team was aiming for was to allow people to use skills as much as possible, as that is the most satisfying way of taking out the baddies. Obviously this is somewhat limited by the resource systems set in place, but since the Barbarian, Monk and Demon Hunter all have resource generating skills, and the Witch Doctor and Wizard both have low to zero cost skills, it's obvious that Blizzard wanted people to use skills as their primary attacks, and not just swinging a weapon.
However, there is nothing stopping you from using a standard attack. Given that we've not seen content past Act 1 normal, there may be situations where killing multiple enemies at once is a bad choice, and if your default attack is bound to an AOE ability, (e.g. Cleave for the Barbarian, Crippling Wave for the Monk, etc.) you may be left to using a default attack.
Arcane Enchanted - Monsters summon a purple laser beam that slowly spins and deals arcane damage. With multiple of these going, you're going to be navigating a maze to get through it. When they spawn, they indicate which direction they initially face before fully appearing.
Avenger - Only champions can have this. Whenever you kill one, the others grow in size and hit harder. The last one left will hit the hardest of all.
Desecrator - Enemies create a fiery (in appearance) zone of damage underneath player characters that deals physical damage over time. Damage starts low but ramps up quickly.
Electrified - Enemies spew out lightning in all directions whenever they take damage.
Extra Health - Self-explanatory. The bonus health is significant.
Fast - Self-explanatory. The monsters are considerably faster.
Fire Chains - This appears on champions only. When close enough together, there are chains of fire linking the champions to each other that deal damage over time to players who stand in the way.
Frozen - Enemies will spawn frozen orbs that spin for a while, damaging and slowing players, and then explode. The explosion deals cold damage and freezes you in place for a few seconds.
Frozen Pulse - An orb of ice pulsates and moves around the battlefield, slowing and damaging players near it. After a while it will explode, dealing extra damage.
Health Link - When attacking a champion with this trait, the other champions nearby will take part of the damage. This makes it tough to take down groups, but has no affect if one is isolated.
Horde - This can only appear on rare enemies. They get extra minions.
Illusionist - Enemies can create 4 clones of themselves. The clones have severely lowered damage and health.
Jailer - The monsters can imprison players so they take some damage and can't move for a few seconds.
Knockback - The monsters can knock back the hero a significant distance (a little over one screen length). The player is slowed significantly after the knockback effect for a few seconds.
Missile Dampening - This one seems to be rather rarer than the rest. It can only appear on rare enemies. The leader (but not the minions) constantly has a rather large bubble around itself that moves to remain centered on the enemy. The speed of all ranged projectiles that enter the bubble are greatly decreased, although if they do make it all the way to the target they will still inflict full damage.
Molten - Enemies leave a trail of fire where they walk that deals fire damage over time to players who stand on it. They also explode a few seconds after they die, dealing damage in an area around their corpse.
Mortar - Enemies gain a ranged attack that fires a volley of a few lobbing bombs every couple seconds. There is a minimum attack distance for the mortars, so standing right near the enemy will protect you from this attribute. This is very dangerous for ranged characters.
Nightmarish - Enemies that hit you will cause you to flee for a few seconds.
Orbiter - Enemies create a central electric sphere, with smaller electric spheres orbiting slowly around it at different distances away. All said spheres deal damage if they touch you. This covers a very large area, but none of the sources of damage are that difficult to dodge.
Plagued - Enemies can create giant pools of poison that deal damage over time to players. These pools are quite large and last a long time and can easily cover the entire floorspace of a room.
Poison Enchanted - Marks an area on the ground as poison, which then spreads in 4 directions diagonally and lasts for a few seconds. Standing in the poison deals damage, but it's avoidable if you pay attention.
Reflects Damage - Dealing damage to the enemies deals a bit back to yourself. A static amount of damage is reflected each time you hit the target. It is only active for short periods of time during the fight. There is a noticeable graphic change associated with the Reflect Damage effect while active.
Shielding - Enemies can turn invulnerable for a few seconds at a time. The effect is fairly frequent, with about a 50% uptime of invulnerability.
Teleporter - Enemies can teleport.
Thunderstorm - Enemies create lightning strikes that also emit small waves of electricity.
Vampiric - Enemies gain life when they deal damage. This also counts damage from other attributes. For example, if you stand on the plague zone from a plagued enemy who is also vampiric, they will gain health.
Vortex - Enemies can suck you next to them and damage you from more than a screen length away.
Waller - Enemies can create walls to prevent escape. They are very good at placing the walls, and will often barricade you in with them. Many ranged abilities cannot go over the walls, even things that appear like they should. For example, a Witch Doctor's Firebomb cannot go over the wall. It is worth noting that rares and champions place their walls differently. Rares will never give their minions the waller ability, but their walls will create long hallways with a blocked end, essentially caging you in with the rare. Champions each can only create one wall each, which has the possibility to trap you in a much smaller area because the layout is not predetermined.
Wormhole - Enemies can create two portals. After a short time, players at either location will be teleported to the other location.
Best Answer
Some monster types do indeed launch meteors at you.
Skills I've seen monsters use: