I just installed a new hdd in my system, and I'm running a full format on it now. What happens if there are any bad sectors? Will it mark them as unusable, or will it just tell me there are bad sectors?
Your hard drive has bad sectors from the factory. They are typically already marked, formatting it (depending on the process used) will just clear any existing partition table, but leave he sectors in tact, ready to be overwritten. If you nuke it (write zeroes to entire drive) it will take a long time, but your bad sector count will remain unchanged.
Drive's firmware will have a set number of attempts per bit during a low-level format, which is much more intensive. If it fails all those attempts the area will still be marked as a bad sector and left out of any partition.
A quick format will wipe only the tables leaving the data intact, but it will lose the knowledge about where particular data is stored.
A full format checks the drive for errors as it goes while creating the partition.