An insinuation is a sly way of saying something, usually something insulting. It can also be a way of worming your way into a group or situation.
"You're dumb!" is an insult. An insinuation is different: it's a way of indirectly saying something. Insinuation evolved from the Latin insinuationem, meaning "entrance through a narrow way.” So an insinuation is like an insult that sneaks in the back door. Mentioning that your cousin could try harder in school could be an insinuation that your cousin’s lazy. Another kind of insinuation is when someone, often through flattery, gets herself accepted by others. Both kinds of insinuations are sneaky.