The stack of stuff on the left is your toilet fill valve. Under normal operation it should allow water to enter until filled to the right level, then shut off. The tub part is probably the float. When the water reaches the right level it should be pushed up and shut off the water.
I see water coming from the side of the unit up high. That's not how most units should operate. The water usually enters the tank low, and a small amount flows through the tube to the overflow pipe. So I suspect a leak or part failure. Different units are made differently, so there's no one failure that I can guess has happened. While I might try to take it apart to make sure it's not something that can be easily fixed, it is probably easier to replace.
For most toilets, you can replace the fill valve assembly without too much trouble. Hardware stores stock different types. I expect it would go easiest if you had a couple of adjustable wrenches, a bucket, and some towels. As long as you can shut off water to the toilet, the job should only take an hour or so for someone that hasn't done it before, but is patient and follows the directions.
Directions will be on the packaging, or you could look up "replace fill valve assembly" to find several videos on it.
The only thing that would worry me is if the tank is non-standard and the $14 Box Store kit won't fit. I've replaced several over the last few years with no issues. But I have one toilet in my house that I know is weird (fits into an odd space) and I would expect takes non-standard parts. If you get into that situation, you might be more limited in finding parts. (Try to find exact model on manufacturer web site or take pictures and go to a plumbing supply store).