There are only three AKC registry accepted coat types Long, smooth, and wire. However this article will cover all four coat types possible.
Long Hair
Traditionally longs were only bred to another long and resulted in true LONG coats such as these two:
However with time (I won’t get into the politics) some breeders began breeding smooths to longs resulting in what is commonly reffered to as a broken or medium long where the hair is only long on the ears, and tail yet still fluffier than a smooth like Pockets pictured here:

While the true long hair dachshund is glorious in its own right, the broken long is easier for the common family to upkeep and much less likely to get tangles and prickers so are popular with families who like the long look and personality but don’t have the extra time to care for a true long coat.
Long hairs tend to be calmer and quieter than their smooth counterparts however there are exceptions to that rule.
Smooth
The smooth hair is an easy keeper that requires almost no brushing only occasional baths or wipe offs. Smooth coats tend to be more active and boisterous so make excellent family and hiking companions.

Wire
Wirehairs are the class clowns they feed on attention and seem to learn new tricks quicker than the other coat types ( I don’t have a clue how coat type affects intelligence). Wire coats do not shed they require stripping which is not as awful as it sounds. Once a week you just gently tug at the long wire guard hairs or run a comb against the hair growth direction while petting your dog and any loose ones will just come off painlessly. Occasionally the hairs that fan out in front of a wire hairs eyes or beard hairs that overgrow the mouth can and should be trimmed. Wirehairs should never be shaved it ruins there coat and makes them more susceptible to cold. Here is Ann in her unstripped coat.

a wire hair with a properly stripped coat should resemble a smooth all but the beard and eyebrows.

Soft/Silky Wire
This coat is a fairly new emergence that comes from the crossing of a long hair on a wire. I have never personally owned one of these coat types and am told they require the most upkeep of any coat type. Some people choose to keep their soft wires natural which takes often brushing to keep tangle-free, and I gave known a few owners who get their soft wires the schnauzer cut. This is not a hair type we purposely strive for but because our wire females do carry the long hair gene even bred to our smooth male they could produce it.

No matter the coat type there is always someone who loves each type.