I have a couple of these https://www.sparkfun.com/products/10213 MOSFETs I use an arduino to generate the 0-5 volt PWM signal and use 12 volts through it to power a 1 amp motor. What exactly would happen if I didn't use the Arduino an put an input of 0-12 volts on the signal line of the MOSFET? Would the MOSFET Just max out the output voltage when the signal voltage reaches 5 volts or will this damage the MOSFET?
It depends on which of the two 'FETs you are using. The page you provided has two data sheets for very similar (but not identical) 'FETs. One of the two has a 20V max gate-source rating, so you'd be OK putting 12V to it's gate, but the other only has a +10/-8V rating. Put 12V on that and you will probably damage it. That's not absolute, but it's beyond what the manufacturer specifies.
Specifically one is +10 to -8V abs max rating for Vgs, and the other is +/-20V rated.
This means don't go near those voltages, its skating on thin ice, since gate overvoltage failure is by catastophic oxide-punch-through. I would trust one of the FETs to handle 12V, the other about 6V nice a reliably and with plenty of noise-tolerance.
You can view the gate electrode as being connected to a capacitor to the source and drain, and this capacitor's breakdown voltage is the abs max Vgs spec. The charge on this capacitor also "magically" controls the channel resistance between source and drain.
Capacitors in practice are marked with "safe working voltages" rather than "absolute maximum", and manufacturers decided to leave us to guess the "safe working voltage" for Vgs in MOSFETs, but it will definitely be less than the absolute maximum which is typically a non-repetitive maximum stress rating.
Thank you, These responses help.