Ratchet freak gives already most of the reasons. I might add that on a grass field the taildragger is easier to handle. The relatively high drag and long lever arm of the tailwheel or skid will help to pull the aircraft straight at low speeds when rudder control power is low. On a hard surface, however, the tricycle gear is easier to handle. Also, not only the weight, but also the drag of two wheels is less than that of three.
If you look at modern airplane designs, those where weight must be kept low (like aerobatic planes) still have a taildragger gear. It is also used in gliders and motor gliders, because they will operate mostly from grass strips and do not want to include internal space to accommodate three retractable wheels.
Big taildragger designs from the early days vanished with the emergence of concrete runways from the 1930s on and the need to brake more during rollout when wing loadings and landing speeds went up. On concrete runways, the tricycle gear handles better and gives much better visibility. Without the ability to brake hard, the rollout lengths of taildraggers would prohibit todays wing loadings. With retractable gears the drag disadvantage is no more, and the increased weight can be tolerated.