Because I've seen lots of people get these words mixed up recently, here's a handy guide to cut out and keep, that will hopefully explain how they work:
ARC means 'curved path'.
It can also be used as a metaphor for a story that has a distinct beginning, middle and end. Like a thrown ball, it curves up then curves down again - in an arc, in other words. 'CHARACTER ARC', 'STORY ARC'.
ARCH means 'part of a building that has two columns either side and a curved part connnecting them at the top'.
A 'character arch' would be an intelligent, talking architectural feature with its own personality.
ARCH- as a compound element means 'chief' or 'superior' or 'more important'.
An archbishop is in charge of other bishops. An arch-enemy is more dangerous than a normal enemy. Your arch-nemesis will hunt you down when your other nemeses have given up the chase. You can fit an infinite number of angels on the head of a pin, but less than a dozen archangels.
The first two meanings (arc and arch) both derive from Latin 'arcus' meaning a bow (that's bow as in '--and arrows'). The second (arch-) comes from Greek 'archos' meaning chief.
Arc is pronounced with a hard 'c': /a:rk/
Arch is pronounced with a 'ch' as in 'chair': /a:rtʃ/
Arch- is normally pronounced in English as 'arch', but in some words taken straight from Greek (such as 'archangel') is pronounced as 'arc'. /a:rtʃbiʃəp/ /a:rkeindʒel/