What type of material is my countertop made of?
I could be wrong, but from the photos, you seem to have a quartz countertop.  

Some brand names of quarts countertops are Silestone, or Caesarstone.  

As good as your photos are, you may wish to compare visually with a countertop distributor to see if you agree with the guess.

---- now that people tend to think it is quartz ----

Quartz is strong (about as strong as granite) but that also makes it brittle.  The problem with brittle counter tops is they chip with a hard enough impact.

There are products to clean and fill the chip.  A quartz counter top installer probably has access to them, and will hopefully have enough experience with them to match color and embedments close enough to not notice.  Another option is to have the edge ground down with a bullnose.

There is no special cleaning for quartz, other than the standard "don't use highly abrasive materials" which will scratch off the polish layer.  Your ring looks like a rust ring, and it may have embedded into the resin binding.  If you call a pro out to fix the edge, you can might ask them if they can buff the surface and re-polish to match.

I would still use trivets and hot pads for hot items.  Sure the counter can take the heat without them, but you want to minimize (like granite) the possibility of thermal expansion of the counter top.  They also provide a padding for setting things down (which can minimize chipping if you slightly drop things on the counter)  While heat expansion won't be very much, the hardness of the material will be such that small amounts of heat expansion could promote cracking.