OK so I am a programmer and not an eye doctor, but between dating an optometrist and a recent experience of getting glasses made me realize that I've learned quite a bit about this stuff. So here's my completely uneducated, ignorant, socially irresponsible and quite likely dangerous take on what different stuff mean.
We all know that 20/20 means you've got perfect vision, except it doesn't. 20/X roughly means that you can see a letter of size X from 20 feet away. So 20/15 is better than 20/20 because it means you can see a smaller letter at that distance. The lowest line on the common vision chart is 20/10 but usually nobody tests that far. Young healthy people are expected to see 20/15, but 20/20 is a nicer number, I guess.
Apparently these are bullshit terms, based on the assumption that if your near vision is poor, you can see well at distance, and vice versa. These terms actually stand in place of several eye conditions, including myopia, hyperopia, and presbyopia. I'll cover these in a minute.
When you're looking at something close, your eye needs all the power it can get. When the natural power of the eye (scientific term here!) is not enough for you to see what you're looking at, your eye tries to help it by forcing some lens in your eye to bend. The bending of the lens gives you more zooming power. This is called accommodation.
As you get older, the lens is not as flexible and you lose your power of accommodation. The loss of near vision due to inability to accommodate is called presbyopia. This is one of the forms of "farsightedness." Presby is a root meaning "old age" and presbyopia is pretty much inevitable as one age. You will need reading glasses at some point in your life, and presbyopia is probably the reason.
Hyperopia is the other condition that falls under "farsightedness." It has to do with how the cornea & the lens focus the image on your retina. Hyperopes do not have this image focused in the optimal way, making the image blurry. Young people can use accommodation to make up for the hyperopia, which is called being a "latent hyperope" as opposed to being a "manifest hyperope."
Myopia is when your eye has too much power, giving you good close-up vision but not allowing your eye to see at distance. It's cause is sort of the opposite of hyperopia. But while hyperopes can use accommodation to increase the power of their eye, there's no way for myopes to decrease the power of the eye.
A diopter is a unit of measurement for lens power. As such, eyeglass prescriptions are expressed in diopters. For example, my hyperopia requires +2 diopters, increasing the power of my eye. Presbyopes also need + diapters. Myopes, whose eyes are too strong, need - diopters. Diopters are expressed as (1 / distance to object). So to see an object at 1/2 meter away, a normal eye needs to accommodate 2 diopters. But if your eye is 2 diopters hyperopic, then you need to accomodate 4 diopters to see it! Wearing a +2 diopter prescription makes up for the hyperopia, allowing one to see the object with only 2 diopters of accommodation.
Astigmatism is when different parts of your eye have different powers, requiring you to wear a toric lens, whatever that is.
Despite sounding like a skin disease, stereopsis is actually a good things. It means you have binocular vision - your brain processes input from both eyes at all times - which helps depth perception. You may not have stereopsis - I don't - if your eyes differ in power. It seems that my vision system uses one eye for near stuff, and one for far.