How many Relics does it take to hatch a dragon?

You need 15 Relics of the same dragon to hatch it immediately (= bring it to 1★, its starting rank). Once hatched, continuing to accumulate its Relics raises its Star Rank.

A useful detail: Relics often drop as "eggs" of 5, so 15 Relics = 3 eggs for a hatch.

After hatching, the star tiers cost (in that dragon's Relics): 40 (2★), 100 (3★), 180 (4★)… up to 800 (7★). All of it is permanent. Hatching is therefore the very first milestone: go for 15 Relics of a dragon you actually intend to play rather than scattering them.