A brown ring on the ceiling is a sign water got in. The cosmetic fix is easy, but it's pointless until you find and stop the source — otherwise the stain comes back.
Find the source first
Water travels, so the stain may be far from the leak. Common sources are a roof leak, a plumbing leak from above, an overflowing tub or toilet, or condensation. Check the attic or the room above during or after rain or water use to trace it. Fix the leak before touching the stain.
Dry and assess
Let the area dry fully. Press on the drywall — if it's soft, crumbling, or sagging, that section is damaged and needs cutting out and patching rather than just painting over. Watch for mould, which needs cleaning.
Seal and repaint
A water stain will bleed through ordinary paint, so seal it first with a stain-blocking primer (a shellac or oil-based stain blocker). Once sealed, repaint the ceiling to match. Skipping the stain blocker is why painted-over water stains reappear.