What do the extra words next to players name and level mean?

Solution 1:

Those words are your character's title. The title is taken from the classmod you have equipped. There are over 100 different titles possible, so instead of writing out the whole list here, I will link to the lists of class mods on the Borderlands 2 wiki.

Commando

Siren

Gunzerker

Assassin

Mechromancer

Under Possible Skill Boost on those pages, the word in parentheses goes before the name of the classmod, so the title of a Commando with the first mod and variant on the list would be Front Line Engineer. This title replaces your normal class name.

Solution 2:

This is your class specialization. When you pick up a class mod and equip it, it'll change your class name from whatever it was to what the class mod is tuned for. Such that my Zero character picking up a Killer class mod changes my class from Assassin to Killer.