Something quite strange has happened with the from the Serial Monitor. For some reason these are the values being read even though according to the code it should be '0' because there is nothing even plugged into the analogue INPUT yet.
This leads me to wonder if my board itself is broken as the code has previously worked for me many times before.
This is exactly the reason...
When nothing is connected to the pin, it becomes an antenna for all electromagnetic interference around.
You need to connect it to the signal under investigation.
For testing, you can connect it to the GND, for example, and measure the values
Read about the concepts of SINK and SOURCE in electronic circuits.
AI's (and DI's for that matter) are high impedance so as to not load down the input signal. Coding pinMode(anyDpin, INPUT_PULLUP) in your setup() code will supply an internal pullup so you can read a switch closed to gnd with no other components.
The AI's need to have an input, or as has been said, they'll pick up any rf noise instead of signal. Put 10k to gnd or +V and you should read within a couple of counts of zero or full-scale. Hook either end of a 5k pot (common in most kits) to +V and gnd, and connect the wiper to any AI, and you should see the reading smoothly track with the pot position. With the right scaling, a readout in absolute volts, or % full scale, is easy to compute.
Not only can you do this for a digital pin, you can also do it for an analogue pin. This will then give you a reading of 1023 for any unconnected pin.
I would only do this for a test unless you want to do something like have a light dependant resistor connected between input and ground to give you readings. But most of the time in this situation you need a more specific value of pull up resistor.
