"Weight/resistance training" is also called strength training. Exercises like push-ups are weight/resistance training. Yoga poses like the plank position are also. So are the exercises done on the weight machines at the gym or with free weights at home or the gym. The focus is to build muscles (as compared to aerobic exercise, in which the focus is to increase endurance and aerobic capacity).
By building muscles, we increase our lean body mass, which is the part that uses calories, so we can eat enough to get the nutrition we need without putting on fat. I also find that I feel better about myself as I get stronger. (Don't worry - you won't start looking like a football player. As we women build muscles, they are leaner than men's.)
Hope this helps!