A cataract is when the natural lens of the eye itself clouds over. It is not a growth in your eye that they can operate on and you'll see the way you did before, the problem is that the lens itself is defective and must be removed. This can happen from age, as a birth defect, trauma, or from some other cause that is not understood. I developed cataracts when I was 25 years old. I am 29 now, and finally had my lenses removed about 5 months ago because I was almost completely blind.
The lens provides about 16 diopters of focusing power, (this can vary from person to person) and all accomodation. (The ability to adjust focus between near and far objects.) As an example, if you wear contacts and your prescription is -4.00 for example, it means that the shape of your eye requires 12 diopters of focusing power, your lens is adding 16 (which is too much) so you need a contact to subtract 4 diopters.
If someones lens is removed, they are not blind, just very very farsighted. (It depends on their prescription prior to cataract, and it also depends on how strong their lens is.) For example, mine was 26 diopters. If I was aphakic, I would be 26 diopters farsighted, and I should be correctable to 20/20 if I wore a contact with a +26.00 prescription in it. Glasses for 26 diopters would be as thick as coke bottles, (with a large amount of distortion) and standard contacts don't come in prescriptions that strong. Which is why they make silsoft.
Usually, they will surgically implant a new lens. (which is made of acrylic or silicone for the most part) This new lens is called an IOL, and is designed to be permanent. However, sometimes for whatever reason they can't. For example, if an infant has cataracts, they have no idea what strength of a lens to use. And its going to change because their eyes will grow. So they give them contacts instead. If there are other eye disorders, it may not be safe to implant and IOL, or if there was trauma to the eye and there is nothing to attach the new lens to, they may need to use contacts as well.
All in all, you don't need a lens. You are correctable to perfect vision without one. (with a strong enough prescription.) The only problem is that the natural lens can change shape. When you look at something close up, it gets rounder, and the prescription gets stronger. When you look far away, it gets flatter and the prescription gets weaker. For example, if you look far away, your lens may be a +16. But if you're reading, it will change shape to become a +20, which brings near objects into focus. A contact lens (or an IOL) cannot change shape. It only has one prescription strength. So if someone is wearing silsoft contacts, they can have 20/20 vision, but only at one distance. They will be unable to read without additional reading glasses. Intermediate distance (like computer) will also be blurry. So normally, they're wearing these very thick contacts, plus progressive glasses, trifocals, or at least a pair of reading glasses. Basically they will have vision like your grandparents, no matter how old they are.
Aphakia is absence of the lens. If you have a natural lens, you are phakic. If you have an IOL, you are pseudophakic. (I am pseudophakic. They miscalculated me a little bit, so I wear normal contact lenses now I think they're air optix.) If you have nothing, you are aphakic. Hence aphakic contacts. That was my ramble.