In a party, the shaman fulfills the following roles:
1) Support and utility (with totems, mostly)
2) Extra melee and spell damage
3) Secondary healer, or sometimes primary healer (though the second is a stretch)
Now for more in-depth analysis.
1) Support and utility
This is where our totems come in. When the group enters combat, the first thing a shaman typically does is drop totems. Totems such as Stoneskin, Strength of the Earth, Healing Stream, and Flametongue are always appreciated by the group. They affect the entire party in a positive way, increasing damage or enhancing survivability. We also cure disease and poison, and have a few other abilities that help the whole group. We can't Purge our buddies, sadly, but we can purge nasty things like damage shields from our enemies. Also in this category would be our anti-spellcaster abilities: Grounding Totem, Earth Shock, and Purge. Not a lot, but it's a good niche and more than other classes get.
2) Extra melee and spell damage
Don't get me wrong, shamen aren't rogues or mages. We're probably not going to be in danger of pulling agro off the warrior unless he's a bonehead. That said, the damage output of a shaman is considerable. We deal nice damage with our weapon buffs and our Shocks, and some of our Fire totems are good for this also. Sadly our most efficient damage spell, Lightning Shield, isn't very useful in a group context, but we have a lot of other options.
3) Secondary or (sometimes) primary healer
This is a biggie. There are never enough healers, so the shaman is often called upon to help out. A shaman really shines as a secondary healer, especially when combined with some of our totems. Healing Stream and Stoneskin totems drastically reduce the damage the party takes while generating very little/no aggro. Our heals aren't as efficient as a Priest or Druid's, which is why it's better to have one of them as a primary. But because of our melee abilities and the efficiency of our totems, we can contribute quite a bit to the group while saving mana for emergency heals. If you want to be a primary healer as a shaman, you'd better spec your talents for it. It's a tough road to hoe. It demands a good group. We're tougher than a priest if we get too much healing aggro, but we have no abilities to get the mobs off of us. (Earthbind totem is the closest I can think of.) The point is that between our heals and our rez, groups love having us around when things go sour. We can save the group when it would have otherwise wiped.
Now, I don't know how any of this compares to the bard, but that's how it is in WoW.