Start with the color and shape
Green, brown, or reddish-brown blotches under where leaves or acorns sat: organic tannin staining.
Rust-colored, uniform, often worse near returns or after a big fill: iron.
Blue-green or teal, sometimes with a metallic sheen, often on a pool with a heater or a history of low pH: copper.
Purple to black smudging: usually manganese, occasionally copper cyanurate.
Chalky white, crusty, or a hard band at the waterline: calcium scale, not a stain at all.
The two tests that settle it
Hold a chlorine tablet against the stain for thirty seconds. If it lightens, the stain is organic and chlorine-based treatment plus brushing will handle it.
Rub a vitamin C tablet on it. If it lightens, it is metal — ascorbic acid pulls metal stains and does nothing to organics.
If neither works, you are likely looking at scale or at a surface defect rather than a deposit, and abrasive or acid work is a different conversation.
Treating each one
Organic: shock, brush, and keep chlorine at a proper level relative to CYA. Prevention is removing debris before it sits.
Metal: sequester first, then treat with an ascorbic-acid-based product, then hold a sequestrant to keep the metal in solution. Skipping the sequestrant after treatment guarantees the stain comes back.
Scale: a chemistry problem before it is a cleaning problem. Get LSI under control or the scale returns, and heavy calcium removal on tile and natural stone is specialist work.
What makes stains permanent
Acid-washing an organic stain that only needed chlorine. You just removed plaster for nothing.
Shocking a pool full of dissolved metal. Chlorine oxidizes it straight onto the surface.
Waiting. Fresh stains lift easily; a stain that has been under water for two seasons is a much bigger job.
