@independentpen The eye doc should've explained things to you when you asked. Find a different one for next time. Eye exams every couple of years or so are really important. I hope this doesn't discourage you.
Anyway, the eyeball thing is called a tonometer and is used to measure eye pressure. The thing very lightly touches your eye for a fraction of a second. I find it helps to focus on the wall behind the examiner while that blue light gets closer.
Measuring pressure is super important as high pressure can indicate increased risk for glaucoma -- a blinding disease. You can't feel high pressure, so it can only be discovered by measuring.
Given that, there are other devices that can be used for people who can't tolerate the tonometer. There is a handheld version used for kids that is less intimidating and can be used on adults. Also, many place have a device that sends a puff of air onto your eyeball and it measure how much your eyeball responds to the puff -- and this somehow measure your pressure (it is the least accurate of the devices, but can still indicate if there is a problem. The tonometer is the most accurate).
The other important thing in an exam is getting a good look at your retina (the back surface of the eye) as that's where your optic nerve lives. This involves a good look using a very bright light -- the light can be uncomfortable for people with light sensitivity. Some places take a photo of the back of your eye using a big machine you stare into.
The back of the eye exam is important a) because there are all sorts of possible retinal conditions that can lead to vision loss, and b) not all glaucoma is caused by high pressures, and if glaucoma is present there will be some damage to the back of the eye.
Maybe write the clinic and say you were unhappy about his bedside manner.