Cloth diapers: Prefolds or fitteds and covers

One cloth diaper option is using either prefolds or fitteds with covers.

Prefolds are rectangular diapers that have more absorbency in the middle and less layers along the sides. (Flat diapers are one-layer diapers that don't have the thick middle.) With either one, you would use either diaper pins or a Snappi to fasten the diaper. There are several folds to use to get the best fit for your baby.
Fitted diapers are diapers that are cut and sewn into a shape that requires no folding, they resemble a diaper and some even have either velcro or snap closures. These types of diapers can be made out of several types of material. Hemp, Indian or Chinese cotton, or bamboo are some options. With the cotton, you can choose natural, bleached, dyed or even patterned. I don't, nor can I find many who do, recommend using the brand you can get from many grocery stores simply because they are not as absorbent or high quality as the others I have listed here.
No matter which you choose, unless you buy used, you'll have to do a number of pre-washes before use to strip the natural oils from the fabric and 'quilt' them up. The more you wash them, the more absorbent they become.
All three kinds would require a cover to make them waterproof.
Covers are generally waterproof or water resistant. PUL (Polyurethane Laminate) is fairly waterproof, but will wick or leak if the diaper gets too wet. Fleece is another option but will wick if compressed. Wool is wonderful because it's breathable, absorbent, and easy to care for.
These covers are either closed by velcro or snaps, and some can be pulled on, reminiscent of the old style plastic pants.

When choosing this type of diapering system, you will need less covers than you will need inner diapers (prefolds or fitteds) because you will be able to reuse the covers without washing as long as no feces get on them. The inner diaper will catch it most of the time, and while the cover may get wet with urine, you would simply set the just used cover aside to air dry, use a new cover in the meantime and swap them out as needed. How many you will need varies depending on the age of your baby and how often you wish to wash. For a newborn, washing every 24-36 hours I recommend at least 2 dozen inner diapers and 6 covers.