Symptom
Fuzzy mould
Fuzzy growth is a fungus or water mould in fruiting mode. Grey fur on fruit is botrytis; grey-purple down under leaves is downy mildew; white fluff on compost is often harmless or damping off.
Where the fuzz is tells you most: on fruit, under leaves, or on the compost.
Likely causes
Ordered from the usual answer to the less likely one. Each links to the full problem.