The ability to Appraise an item boils down to a Single Intelligence (INT) check.
Having a proficiency in a skill or tool that would grant one knowledge pertaining to the item allows one to add their proficiency bonus to the Check. See below for a few use cases:
Arcana - magic items, spell scrolls, potions (as long as they are identified fully)
Religion, History - art (depending on the subject), religious icons, replica's of legendary items.
Smith's Tools - Weaponry, Armor
Woodcarver's tools - Chests, Wooden objects
Jeweler's Tools - Jewelry, Gems
Example DC's
Common: DC 5
Uncommon: DC 10
Rare: DC 15
Super Rare: DC 17
Very Rare: DC 20
Legendary: DC 25
Epic: DC 30
On a Failure, there is a chance the appraisal will present a lower value, a higher value, or the inability to determine the value. Depending on DC and how far off the "Passing Result" the roll is.
A character may only attempt to appraise any specific item one time.
On a Failure, see "Forcing a Check" If additional party member wish to provide a Second Opinion.