Weight training, weight lifting, power lifting, and bodybuilding are all similar because they share a common goal: to create a physical challenge and improve someone's health (either physically or mentally).
However, although they are similar and focus around body power, different training approaches can be used.
Weight training: can include dieting, changing sleep schedule, and exercising consistently.
Weightlifting: using dumbells, weights, bars, etc, to achieve muscles in parts of the body such as arms or back
Powerlifting: Similar to weight lifting, just more extreme.
Body building: Creating a muscular and fit body, by weight/power lifting, eating healthy, and working out.