The creatures most correctly called daddy-longlegs are in their own separate Order which is Opiliones. Common names for this Order are 1) daddy-longlegs, 2) harvestmen and 3) opilionids. They are characterized by having one basic body segment which shows segmentation on the posterior portion, at most 2 eyes and all 8 legs attach to the pill-like body segment. They are usually found under logs and rocks, prefer moist habitat although they can be found in the desert, often have long flexible legs (in the temperate Northern hemisphere but there are also short-legged daddy-longlegs) and they do not produce silk so therefore they are never found in webs unless they are being eaten by spiders. Because they are found under logs and other stuff which people most often are not turning over, most folks don't run into daddy-longlegs very often.
Another creature often called daddy-longlegs are actually spiders. These long-legged spiders are in the family Pholcidae. Previously the common name of this family was the cellar spiders but arachnologists have also given them the moniker of "daddy-longlegs spiders" because of the confusion generated by the general public. Because these arachnids are spiders, they have 2 body basic body parts (cephalothorax and abdomen), have 8 eyes most often clumped together in the front of the body, the abdomen shows no evidence of segmentation, have 8 legs all attached to the front most body part (the cephalothorax) and make webs out of silk. This is most probably the animal to which people refer when they tell the tale because these spiders are plentiful especially in cellars (hence their common name) and are commonly seen by the general public. The most common pholcid spiders found in U.S. homes are both European immigrants. Pholcus phalangioides is a uniformly grey spider with rectangular, elongate abdomen and is found throughout the U.S. Holocnemus pluchei also has a rectangular, elongate abdomen but has a brown stripe on the ventral side (the belly side - which is typically directed upwards since the spider hangs upside down in its web) which covers its sternum and is a stripe on the abdomen. These spiders are very common along the Pacific Coast. and into the southwest deserts.