I have a kind of vinyl curtain for a sliding glass door and the curtain rod broke off. What are these kind of "curtains" called?
That is a vertical blind, but not all are built the same. You need to try to find the manufacturer. The part that is broken is the rod holder. The picture looks like it is a "snap in" part. However the problem is sometimes with the older units the plastic breaks easily and trying to install a new plastic part results in other parts breaking.
There are companies that will come to your home to do a repair, but most of the time, the cost is greater than purchasing a new blind. If you know the age and it is older that about 7 years, you should consider a new unit. If the slats are in good condition you can also purchase a new "Head Rail". The slats are easily removable from the plastic holders. That would be less expensive than an entire blind.