Admissions cares about making a positive prediction of your success and whether you will make proper use of resources, along with contributing in the future. No one thing lets people make that prediction, so many, many things are taken in to consideration.
Positive indicators, including successful internships or research, help you. Negative things, such as poor grades, hurt. Letters of recommendation can be very important.
Other helpful things: Indications of hard work. Indications of working well with others and contributing. Wide interests. Flexibility. But no one thing is absolutely determinative. And individual evaluators will all have their own ideas about the "most important" thing.
So, you want your graduate school application to be filled with positive indicators so that people don't need to make decisions "at the margin" in your case.
But an internship or research experience, alone, won't assure your admission, not will its lack, assure rejection. It is one factor among many.