To cosset is to pamper or spoil. Your mom might cosset her beloved lap dog, feeding him homemade meals and singing him to sleep.
If your parents used to cosset you when you were little, they may have protected you from injury and indulged your every whim. Your temptation might be to cosset small children or elderly relatives, pampering them and shielding them from difficulty. In the 17th century, cosset had the additional sense of "fondle and caress," as you might cuddle a baby or a puppy. Earlier, the word was a noun meaning "a lamb raised as a pet."