When we press the bulb of a dropper with its nozzle kept in water, the air in the dropper escapes in the form of bubbles. Once we release the pressure on the bulb, water gets filled in the dropper. The rise of water in the dropper is driven by:
Atmospheric pressure
When we press the dropper, the number of air molecules decreases which results in the decrease in pressure. But, the outside pressure or atmospheric pressure remains constant. Due to this pressure difference, there is an upward force which pulls the water into the dropper.