When do I use a Mosfet, another transistor, Optocoupler Relay, Solid State Relay, and Solid State Relay?
Also, are there any other types of relays?
I have used "all of the above". It all depends on the current and voltage requirements of the load.
I want to run a 12v/.4 amp magnetic lock and a normal 120v/100w light bulb.
Cost is not an issue. I want the best quality wise for my project.
I had a project where I chose to use 7 solid state relays because I was too lazy to build 7 MOSFET driver circuits...
I don't know what an "Optocoupler Relay" is...
Solid State Relay, and Solid State Relay?
Same thing? 
Relays are electrically isolated. A MOSFET or transistor by-itself is not isolated but if you add an opto-isolator you can make your own solid state relay.* Lethal AC power line voltages MUST be isolated from the Arduino, and from you!
An opto-isolator is a low-power device. A solid state relay can is optically-isolated and it be considered a high-power opto-isolator, and in fact there is probably an opto-isolator inside.
A regular bipolar transistor turns-in with about 0.7V. A MOSFET requires a few volts (or more) to turn fully-on. A MOSFET has lower on-resistance than a regular transistor so it can switch more current while running cooler and MOSFETs are often used in high current applications.
A single MOSFET or transistor only works with DC. Some solid state relays can switch AC. In general, they are not interchangeable. Many (most?) AC solid state relays are built with TRIACs and with DC they will latch-on and won't turn-off until power is removed.
*I've made my own AC solid state relays with a TRIAC and a special TRIAC-driver opto-isolator.
DVDdoug:
I don't know what an "Optocoupler Relay" is...
Same thing?
Basically, yes. It's a relay module with an on-board optocoupler.
DVDdoug:
Many (most?) AC solid state relays are built with TRIACs and with DC they will latch-on and won't turn-off until power is removed.
A TRIAC drops about 0.7V so lots of heat to dissipate at high currents. A possibly more efficient alternative is two n-channel MOSFETs in series, one reversed, sources connected.
It's even possible to switch AC using a MOSFET & bridge rectifier, but then you have two diode drops.