"At the beginning of your upkeep, put a +1/+1 counter on equipped creature if it's white" is a triggered ability of the artifact "Ring of Thune".
It triggers at the beginning of your upkeep, if Ring of Thune is on the battlefield, and it references "equipped creature". If there is no creature currently equipped with Ring of Thune, the ability does nothing. If there is, that and only that creature get a +1/+1 counter.
The +1/+1 counters stay on the creature, they don't belong to the equipment.