Some mobs are especially difficult to defeat, usually requiring more then one player to team up. Those mobs are called 'Elites' and are easy to identify by a golden dragon that surrounds their portraits as well as the 'Elite' description beside their level in the tool tip.
While in some cases, certain classes can solo a particular Elite, it's recommended to bring friends along. There is no way to easily tell if an elite is solo-able, requires 2 people, 5 people or even a whole raid to defeat. It's usually better to play it safe. Most quests that require killing an elite will indicate how many players are recommended.
Elite mobs always have more hp, do more damage and take less damage then similar mobs of their level. They quite often have special abilities as well, making them even more dangerous. Elites in the open world are usually there as part of a quest or simply to make the zone more dangerous and unique. Such elites often have better loot then normal mobs.
The majority of mobs in instances are elites, which explain why those are considered group activities. The vast majority of elites in the open world were removed during patch 3.1 on account of very few players being available in a zone together to tackle these extra-challenging targets.