Samsung. I don't know the model. But it is less than 5 years old.
The model number may be on the front of the machine, the back/bottom of the machine, the system information screen (via the setup menu), the instruction manual, etc.
And being less than 5 years old isn't saying much, especially for a Samsung as they've had certain models of blu-ray players go unsupported after 3 years.
I'm going to assume that means you've never updated your machine. o.0
Blu-ray players are like computers and are updated to support changes in blu-ray disc encryption and new playback features.
It is odd that you could play the first two discs, but even so it is likely that your player cannot properly read some portion or feature of the disc required for proper playback.
Without the model number I can't say exactly how to update your player, but it will be either an update through an ethernet connection, a usb flash drive, or possibly a dvd burned from your computer (that last one is rarer, but a possibility).
So try to find your instruction manual and look up "how to update your blu-ray player".