Mesmer

Summary

Mesmer is a light armor class that uses illusions to deal damage, control enemies and support allies. It also shatters illusions to grant various effects on enemies and allies alike. For weapons, mesmer uses main/off-hand sword, greatsword, staff, main-hand scepter and off-hand pistol, torch and focus. Of light armor classes, mesmer has mediocre health pool. In solo play, illusions also distract the enemies and smartly played mesmer can do very impressive things by having illusions take most of the damage while burning the enemies down.

Mesmers have been long considered a low DPS class, mainly good for CCing enemies (with "Moa signet") and granting distort to allies to help them avoid the most dangerous attacks.

Builds Role
 Condi DPS Mirage DPS

But with latest changes, power DPS mesmer got boosted a lot and at least in PUG-raids it can out-DPS many other classes if played properly (and to be honest, power DPS mesmer has probably the easiest rotation of all classes).

Chronomancer

With Heart of Thorns, mesmer got a new elite spec "Chronomancer" that uses shield and raised the mesmer to be mandatory for any high-DPS squad composition, thanks to high support capability it has in forms of wells and blocks.
Chronomancer's wells grant alacrity and quickness to allies and since keeping up quickness requires using equipment with Concentration-stat and chronomancer has lots of ways to avoid hits, chronomancer also became THE Tank due to least DPS-loss while in tanking gears.
Since mesmer indeed uses illusions for damage while chronomancer uses illusions to keep up alacrity, chronomancer's damage is indeed very low regardless of equipment.
Chronomancer is also famous for it's CC capability thanks to it's 5th shatter, Continuum Split, that restored the chronomancer to it's state 1,5 seconds earlier with all cooldowns, health and position in tact. 
This allows chronomancer to use Time Warp and Moa Signet every 90 seconds (Split's CD) or twice in a row with their original cooldowns. Also, chronomancer has elite skill Gravity Well that also does heavy breakbar damage with 90 seconds cooldown.
 

Mirage

 With Path of Fire, mesmer finally got a DPS elite spec "Mirage" that uses main-hand axe. Mirage introduces some new mechanics, like manually setting a new target for illusions with Axes of Symmetry (axe 3) and Illusionary Ambust utility skill, Mirage Cloak replacing evade with evasion that doesn't move the mirage anywhere and doesn't interrupt any other actions and mirage mirrors that grant mirage cloak when touched, as well as Ambush that gives a special skill to weapon's skill 1 slot after gaining Mirage Cloak.
Trait-wise, Kitty sees Mirage as quite condi-DPS focused specialization.