Let me add a little to YOGI's post. Usually the RAID management software that you used to create the RAID array in the first place will help you if one of the drives fails. But first, yes to your question - the system will continue to function for you by using only the non-failed HDD. You should replace the failed unit and restore the RAID1 array as soon as you can. This is the REAL reason for RAID1. It is NOT an automatic backup!! But in critical applications where you cannot tolerate significant downtime, this will keep you running until you can plan some downtime to make repairs. For example, I use a RAID1 array in a POS computer system to run a retail store sales desk. I don't want it failing in the middle of a day, but I can repair after the store closes at night.
After you have seen the failure message and entered the RAID management screens, usually it will tell you which of the two HDD's has failed. At this point you usually will use its menus to "break" the array into separate HDD's. If it is not clear to you exactly which HDD is faulty, you can disconnect ONE and reboot to see if it still works or not. Then you replace the failed HDD and reboot into the RAID management software again. This time you Rebuild the array, which will copy everything from the good HDD to the new one and re-establish the proper controls and links so you have a fully functional RAID1 array again.