At the bottom of my screen is a purple bar that fills up as I fight, what does this bar do?
That is the Charge Bar. The Charge Bar encourages you to use normal attacks, as it the fastest way for you to charge them and improve the effectiveness of the your Active Skills, where you have to manage your limited mana pool.
The amount of damage that you need to inflict to fill your charge bar scales with your level. If a low DPS weapon for your level will charge slower than a high DPS weapon for your level.
There are Active Skills such as the Embermage's Blazing Pillar (not entirely accurate skill, but good for building charge in grouped up mobs) that builds your charge at a faster rate, and there are skills that don't accumulate any charge (e.g. Magma Mace, Firebombs, Firestorm, and Hailstorm).
Each Charge Bar for each class works in a different manner:
With Berserker, when it's full, you will enter the frenzied state. You will run faster, attack faster, and always crit until the frenzy ends. It's worth noting that the frenzied state countdown doesn't begin until you hit something (after it's been filled up). You could potentially beat up that practice dummy in town to get your frenzied boost and then port into your destination and start the fight with a boost. The charge starts decaying a few seconds after combat and drains at a slower rate than the Outlander's charge bar.
The Outlander's charge bar works a bit differently compared to the other classes. As you fill it up, you will get the passive bonuses (Up to +10% Cast Spd, + 10% Dodge, +10% Crit, +10% Atk Spd) you get. There doesn't seem to be a break point threshold for each +X% bonus gain. The more damage you do, the charge increases, but it's also constantly decaying, so it's pobably the slowest to fill out of all the classes. Filling up the gauge up has no particular effect unlike the other classes. Additionally, it's worth noting that an Outlander with no charge will deal additional damage and stun the next enemy unit they hit. This bonus stacks with bonuses from shotgunnes.
The Engineer's charge bar has specific charge points (up to 5) and that a large number of their abilities use up in order to activate, and/or are more powerful (longer range/more damage) based on how many charges you have. The bar fills and drains much like the Outlanders, but a charge point seems to stay full longer after the charge point has been completely filled.
Embermages are very straight forward, when the gauge fills up, the Embermage goes into a concentration state (for 12 seconds) where all skills cost no Mana and they do 25% more damage on top of any bonuses (tiered, passives, etc.) they've invested in. It's best to save your more powerful mana heavy skills for this state.
That is dependent on character class, but its basically like a charging bar. When you get it to the max it usually makes your character get a buff, like the beserker's frenzy dealing more damage and faster attack speed. Also it should show the exact effect if you hover your mouse over it.
That is your character's Charge Bar, which is filled by attacking or damaging enemies (check the hover-text description for your class, it seems to vary). Normal attacks always seem to add charge and so will most skills as well - a skills' description will explicitly state when that skill doesn't add charge to the bar, so watch out for that.
Note that for all characters the charge bar will eventually deplete if you are not actively filling it.
You should note that the Charge Bar has very different effects for each class, most of which you can see if you read the hover-text by placing the mouse point over the bar, but I shall detail here:
Embermage - When the charge bar is full the 'mage is granted 12 seconds where they can cast skills (and maybe spells?) for zero mana cost and with a +25% damage bonus. After this time is up the bar is left empty, ready to fill again.
Berserker - Operates in a very similar way as the 'mage, once full the Berserker gets a 5 seconds (which can be extended by a passive skill) of guaranteed critical hits, and once that time is up the bar is left empty.
Outlander - As the bar fills the outlander gets a boost to a several different stats (cast speed, critical chance, attack speed and chance to dodge) in proportion to how full the bar is (I think I saw +10% as the cap when full, so +5% at half full, and so on). As far as I could tell, the bar being full confers no additional bonus, however, having an empty bar will make the Outlander's next attack cause additional damage and potentially stun the target.
Engineer - These guys have markers every 20% of the way along the bar (so there are 5), each one representing an available "charge". On their own, they do nothing, but many of the class's skills will consume one or more charges (if available) in exchange for being stronger than if used without a charge. Check individual skill descriptions to see if charges are consumed and what the effect will be when this happens. Having a full bar doesn't seem to provide any direct additional bonus.