Using Arduino for 3d heating bed
The MOSFET doesn't have to dissipate 150 watts. The heating pad does that. The mosfet only needs to flow the 12.5 amps when heater is on, and then block the 12 volts when you want the heater off.
    The Rds (resistance) of the mosfet could typically be 31 mohm (0.031 ohms) when on.  The power that the mosfet would have to dissipate then would be about 4.8 watts.   http://www.mouser.com/ds/2/200/irfz44npbf-222819.pdf

Choose a mosfet with blocking voltage higher than 12 volts. The above mentioned link is for a 55 volt mosfet.  

Regarding protecting your arduino (or other system), there are lots of driver circuits you can find here on EE stack exchange that discuss the matter.

As regards relays, there are many many relays that can be used in DC circuits.