If you (or Ulie) is sitting on a swing, of course she will experience gravity pointing straight down. Because swings are attached by ropes tied to a branch, the tension force points in the direction of the rope. If she is simply sitting there, there will also be the gravity force from Ulie, and a normal force counteracting it.
So, two different gravity forces (one of Ulie, and one of the swing), the tension force of the string, and the normal force on the swing base (where Ulie is sitting). If the swing doesn't break, then the net force of the system is zero.
