Windows obviously formats the partition you choose to install it in (otherwise it wouldn't be able to install itself), but the install program won't format any other partitions unless you specifically ask it to.
If you didn't specifically create a partition for the rest of the space then it will be marked as "unallocated".
A "quick" format writes the volume metadata (empty directories, freespace map, master file table, etc.) and nothing else, so it's very fast. A "full" format actually writes to every sector to verify that its usable and places any sectors that couldn't be used into a "bad block table".