How to make a boost converter circuit
Also as an aside, the thing labelled "L" on the diagram, the inductor,
  what is it's purpose?


If you have an elastic band stretched across an opening and you pull it down in the middle (to ground) then release it, said elastic band responds by rapidly rising and if you have your hand in the way then you can get a slightly painful little recoil slap. Here's a nice picture to consider: -



You can fire the little object attached to the rubber band quite a distance.

Now that analogy isn't precisely what happens with the inductor but it's near enough to make use of.

So, instead of allowing the elastic band to freely recoil (and waste all the energy you have put into it), imagine you mechanically harvested that recoil to push a little mass/object a little bit higher each time you released the elastic band.

You'd need the mechanical equivalent of a diode so that when the elastic band was pulled down you didn't return the object/mass to ground. That wouldn't be too hard to build but you don't need to build it - you just need to understand the analogy.

So, to get 7V from 5V you need to turn on the transistor for long enough to build up a certain current - that current defines the energy stored in the coil and when that energy is released via the diode and into the capacitor and load, it's enough energy to keep it at 7V and the cap is big enough so that the 7V doesn't droop too much when the inductor is being pulled to ground.

However, the difficulty arises when you try and control the output voltage to a precise value - it is highly load dependent and highly input voltage dependent and so normally what happens is that an op-amp control loop defines the duty cycle of the inductor to prevent too big an output voltage being generated.