The typical models cover transistors which are done in optimal way - you can say they are ideal.
Corners like fast fast etc cover usually +/-3sigma range from the typical and are supposed to make sure that your design will work when there is a shift in manufacturing process.
Also it covers use of different fabs (if you use UMC or TSMC your chip could be done in any of ther x fabs unless you specify).
Monte carlo just runs the gausian distribution for some basic parameters of the devices like Vt, L, W etc. and if the models are setup correctly corner models will be at the end of the monte carlo distribution.
What can change? If you have ideal process as you read in books then there would be no need for ie mismatch. In real life nothing is ideal so if your diffusion time is slightly longer, the day before the fab personell had few many you can change Vt or have slight variation of the L or W. It is all statistical stuff an if you want to understand more just google it up.
Should you run the corners? Not if you are highly self confident or don't care(like if it is a school project). Yes if you want to keep your job.
I also recommend to open the model file in text editor and compare Typical model to Fast and Slow - right there you will see what is changed.
Then if you have access to any fab data get the wafer probe data and look at it (I mean process test structures probe not your device!) there you can see lot to lot how things are changing.
It is actually really cool to see how much things vary and yet we still can make ICs!