During a heart heart catheterization you're usually awake but you will be sedated as in they'll make your body feel numb so you won't feel the heart catheter being inserted.
For other heart procedures such as ablation, heart valve repair or heart valve replacement you're usually almost always sedated fully under general anesthesia where you'll be fully asleep and won't be aware of the heart surgery or procedures.
My dad had heart catheterization done a few years ago and it was very safe and he recovered fully after it soon after.