Types of cloth diapers

Flats:  These are the old cloth diapers that most people think of when you say, "cloth diaper."  This is a square piece of flat fabric that can be folded and fastened with pins or a Snappi.
Price:  $
Difficulty: +++++
Dry time:  /
Ick:@@@
Cover required:  Yes 

Prefolds:  These are similar to flat diapers, except that these have a thicker portion in the middle of the diaper.  These can be folded and fastened with pins or a Snappi, or they can be laid flat in a wrap/cover.  There are varieties within this type of diaper including Indian prefolds and Chinese prefolds.  The first is usually a softer cotton, but may wear out more quickly.  The latter is more sturdy in material and stitching, but not as soft.
Price:  $
Difficulty:  ++++ (if fastened) +++ (if laid flat)
Dry time:  //
Ick:  @@@
Cover required:  Yes

Contours:   These diapers have almost an "I" shape, with one end (the back) being wider than the other.  They are wrapped similar to a disposable.  Some may need to be attached with pins or a Snappi, but most will stay put under a diaper wrap.  Contour diapers are usually sized.
Price:  $$
Difficulty:  +++
Dry time:  ///
Ick:  @@@
Cover required:  Yes

Fitteds:  Fitted diapers are exactly what they sound like.  They are fitted, similar to disposables.  Fitted diapers can be sized, or they can be adjustable to fit from birth to potty training.  These diapers can attach by either velcro or plastic snaps.
Price:  $$$
Difficulty:  ++
Dry time:  ///
Ick:  @@
Cover required:  Yes

Pockets:  Pocket diapers can refer to any diaper with a "pocket" for adding doublers for more absorbency, but in this instance, I am using it to refer to diapers that have an outer waterproof layer, an inside thin (usually "stay-dry") layer, and a pocket.  These diapers cannot be used without an insert - the inner layer is not absorbent.  The diapers are stuffed with an insert or doubler, then fastened similar to a fitted diaper with either velcro or plastic snaps.
Price:  $$$
Difficulty:  ++
Dry time:  //
Ick:  @@@@
Cover required:  No

Hybrids:  Hybrid diapers consist of a diaper cover and a disposable insert.  These diapers appeal to many people because only the cover needs to be washed and everything else can be thrown away, flushed, or composted (depending upon the brand).  The inserts do not stick to the diaper; they just lay flat similar to a prefold.
Price:  $$$$
Difficulty:  ++
Dry time:  /
Ick:  @
Cover required:  No (a hybrid diaper IS basically a cover)

All In Ones (AIO):  All in one diapers are exactly what they sound like - one piece.  These cloth diapers are the most similar to disposables.  They can fasten with either velcro or plastic snaps, and they may be sized or be adjustable to fit from birth to potty training.  Many AIOs have pockets for the option to add absorbency.
Price:  $$$
Difficulty:  +
Dry time:  ////
Ick:  @
Cover required:  No

Diaper cover or wrap:  Covers are mostly made of PUL (polyurethane laminated fabric) or wool, and can fasten either by velcro or plastic snaps.  Covers can be sized or they can be adjustable to fit from birth to potty training.

Inserts or doublers:  Inserts are used in pocket diapers or can be used as an added flat layer in other diapers to increase absorbency.