First and foremost, shamans drop totems. These are localized temporary buffs for the immediate 5-man group in a static range around the totem: 20 yards or 30 yards with talents. I'm not going to make everyone remember the names of spells, so I'll just summarize the totem options of each element.
Earth - +strength, fear/charm/sleep dispel, reducing melee damage, slowing enemy movement, taunting/stunning totem, and a summoned earth elemental medium aoe tank.
Air - +agility, extra attack weapon buff for group, absorbing one direct damage spell, +spell damage, +nature resistance, reducing ranged physical damage, reducing threat generated, and a sentry spy that lets you see through it at great distances.
Fire - weapon buff that adds fire damage for group, single target fireballs, aoe pulsing low fire damage, aoe one-shot large fire damage, and a summoned fire elemental that does large aoe damage (but is fairly squishy).
Water - periodic health regen, periodic mana regen, disease cleanse, poison cleanse, and +fire resistance.
And now on to the primary and talent abilities:
Healing potential - Shamans have a short cast lesser heal, and a longer cast bigger heal. The healing stream totem scales with +healing gear. Earth Shield is a 41 point talent in the Restoration tree which can be cast on anyone, healing them instantly when they take damage (only once every few seconds, per the tooltip). And, if they're Draenei, they get the racial heal as well (but then all Draenei get it, too.)
Melee abilities - A shamans melee abilities relies entirely on their "enhancement" talent tree. Using talents, they can equip 2-hand weapons, dual wield, and gain a chance to parry. The 41 point talent in Enhancement is called Shamanistic Rage, which allows them to gain 15% of their attack power as mana via a chance-on-hit weapon proc. Given the right +hit, high attack power, and a little luck, a shaman can go from almost no mana to almost full mana each time this ability is used.
Elemental combat - This tree has become recently popular with the content patch just before The Burning Crusade. Giving major buffs to spell damage, mana regen, and spell crit rating, this tree is very powerful though tends to be an agro hog. The 31 point talent grants an ability that when activated causes the next fire/frost/nature spell to be a guaranteed critical hit. The 41 point talent gives a new totem which gives the group an extra 3% spell critical chance using the fire totem element.
Lesser-known major abilities (the spells that many people may consider exploits, but I assure you they are NOT!):
Self-Resurrection - On a one-hour cooldown, or 40 mins with talents, shamans can resurrect upon dying. It requires an Ankh reagent. Shamans who die while the group is in combat can resurrect, however they will have very little hp/mp and they do not retain their buffs when they stand back up.
Water Breathing - Yes, shamans do have this, but it requires a Fish Scales reagent (unlike the warlock equivalent which has no reagent.)
Water Walking - The most unbelievable buff in the game, which is not as commonly used, is Water Walking. Requiring fish oil as a reagent, shamans can buff anyone to be able to walk on water as if it was solid ground. This effect is removed if the player is struck in combat, strikes another target, casts a damaging spell, shapeshifts (ghostwolf/druids/costumes), or mounts. You can, however, mount a flying mount and immediately take off as you lose the buff when you mount. You can fish while water walking, and you can sit and eat/drink on the surface of the water.
Heroism/Bloodlust - This ability is one of the most powerful in the game, which a shaman only receives at level 70. Heroism increases melee, ranged, and spell casting speed of the entire group by 30% for 40 seconds. This ability is most often used as a boss-killer, and if you have multiple shamans in the same group, you can cast another as soon as the first wears off. Unfortunately, this does not buff an entire raid (which I think it should.)
[Edit: Heroism is the Alliance version of the ability, and Bloodlust is the Horde equivalent.]
I hope this helps clear up some of the mystery of the shaman class, and brings to light some of their lesser-known abilities and skills.
[Edit #2: I have corrected a couple of my numerical errors in this post. Thank you for your comments and criticism, as I will use it in future posts.]