I used to be really confused whenever I had an instructor or instructions which told me: “Take your belly button to your back!” or “Belly in!” or “Engage your core!” I asked my yoga and dance teachers to explain to me what "engaging your core" really is and how it differs from sucking in and flexing.
Sucking in: This action is too drastic. Your whole abdominal region collapses inwards causing your back to round. Also, your shoulders tense and creep towards your ears.
Engaging your core: Imagine that you are zipping up a pair of jeans. You are standing up tall, tailbone is tucked in and not sticking out, and you have a little tension in your abdominal muscles. You are staying flexible in this position.
Flexing your abs: This move is too static. Your breathing is hindered. You are stiff and rigid limiting your ability to perform many poses and exercises.
Engaging your core without overdoing it allows you to develop both balance and strength. This is a silly picture, but I wanted to illustrate to you the differences! Hope it helps >.<