DC Questions and my understanding
Battery x is rated at 5volts, 2500mAH.
  
  Device y draws 5 volts, 5mA. This means that the battery would last 500 hours correct?


Correct.


  Now, device z draws 5 volts, 3000mA. This means the battery would last under an hour correct?


Correct.


  Now circuit d draws 5 volts, 2500mA. The circuit would last 1 hour.


Correct


  Now take a AC -> DC adapter, it is rated at 12 volts, 1A.
  
  What does that amp in this circumstance refer to?


That's the maximum the supply is capable of providing before it overheats and melts.


  Is it to work out the maximum resistance that a device plugged in could have?


No, it's purely a maximum rating.


  Would this adapter be working as the battery in a circuit?


In effect, yes. It can be seen as a fixed voltage source.


  So, batteries don't output amperage. It changes depending on the voltage/resistance of the device.


That is correct.  A battery is a (reasonably) fixed voltage, but the current is dependant on the load (Ohm's Law).  The mAh of a battery is how much charge is stored in it. A power supply doesn't have this concept as the charge is essentially infinite.  A battery, at 2500mAh, has enough charge to supply 2500mA for 1 hour, or for other currents, an approximation:

$$
t = \frac{mAh}{mA}
$$

As @PeterBennet mentions, it's rated at a pre-defined discharge time. Discharging faster than that rated time can result in less apparent capacity.


  Now wattage. Why are some devices rated in watts?
  
  For example my PSU is 600watts.
  
  So it outputs 600watts total over a bunch of 12v, 5 volt and 3 volt cables? 
  It's input is 220V, so the amperage it draws is: 2.7Amps?
  
  Or is that the amperage it outputs?


Again, it's the upper limit.  It can output 600 watts in total over the different outputs before it overheats.  For a supply with multiple voltages it's often easier to specify the total limit using a voltage-agnostic value, like Watts.  If you just specify current, then you have to specify it for each voltage, and if some current may be shared between voltages (e.g., cascading 3.3V off 5V) then it gets more confusing. Simpler just to give a total power rating.