Cone crushers are sort of self explanatory in that the crushing area is conically shaped with a comparable impeller/crushing head. Sort of like a mortar and pestle like compounding pharmacies use.
Impact crushers are hammermill types of crushers. They use a rotor and box crushing area where hammers are rotated on a rotor and material is crushed through the surrounding screens (same principal as an auto shredder).
Jaw crushers or slip jaw crushers, use two articulated heavy jaws that either move side to side or one part is stationary and the other moves, and the material is crushed between the two jaw plates.
They can all be used for aggregate materials, although some work better than others (also personal preference and price enter here). Wood wastes work better through conical (or tub) grinders.
I hope this helps. Let me know if I can help some more.

What types is the best to make 3/4" or 1 1/4" product? Also, what type is the best for mobility: ie-onsite crushing.

They can all be mounted on mobile platforms, but for your app I would think a jaw crusher would be fine. You've got less to mess with, no hammers to change. There are a slew of mobile crusher manufacturers out there. I am personally familiar with Extec, Vecoplan, and Doppstadt. You can probably check around your area with some used supp;iers and pick up one that will serve for onsite use.

Thank you very much for the insight. It was very helpful. One last question, do you own/ operate a crusher? If so, what brand, what type, and are you happy?

No, I currently do not have a crusher, but the last one I used was a leased Extec c-10 jaw crusher (I only needed it for one job). It was sufficient for the job but if I were to buy one I would probably go with a larger unit (just for flexibility of throughput).

In a related note, I have recently signed as an independent rep for MetalTech Systems, a General Kinematics/Sherbrooke OEM distributor. We manufacture custom material handling systems, so if you are looking for a site oriented c&d screening/recycling system, I would like to be able to help.
You can pm me if you want and I will send you some contact info.

Thank you very much.