Good news: when you block someone annoying on Instagram, they won't be notified. Bad news: they'll be able to tell you've blocked them if they go to your profile page. More good news: they're blocked, so they can't annoy you anymore anyways.
Instagram is for more than just sharing your latest travel snaps and shopping hauls. It's also a great place to see what your ex is up to and whether or not your cousin is still dating that super hot guy you met at Christmas. But if you're the ex who's being checked up on and you want to put a stop to it, you can block whoever you want.
When a person is blocked they won't get a notification that you've done it. Once blocked, they won't be able to see your stories or posts, send you messages, or search for your page. If they have your username, they could go directly to your page, but they won't be able to follow you or interact with you at all. They can still mention you (if they know you're username) in comments, but you won't get a notification, so it's like they don't even exist!
To check if you've been blocked — which also works on Twitter — just sign out of your Instagram account and search for the person you think blocked you. It should show up as usual to you if you're not logged into an account they've blocked.