Molecules on the droplet surface don't have neighbors on one side. There is a strong net force from the tug of the water molecules below them, which pulls them towards the center of the droplet.
Surface tension is the work you have to do to increase the area of the drop by a unit amount. Since work has units of force times length, surface tension has units of force per unit length.
A small amount of liquid in a very thin dropper won't run out unless you blow it out- it's held in the tube by the surface tension of the drop on the tip, which exerts an upward force. If there is enough liquid in the tube, though, there will come a point where the weight of the liquid overwhelms upward force, and the drop will fall.
If surface tension is force per unit length, you can crudely predict the weight of that droplet by multiplying the circumference of the dropper tip by the liquid's surface tension. For example,
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