There are three main styles of droppers: bulb and pipette, euro insert, and dasher.

The ones you saw with the rubber bulb usually have the bulb and pipette (tube) attached to a screw cap but there are special styles where the bulb also forms a stopper or the pipette is wide and forms a ground glass joint with the bottle neck.

The euro insert is the type found in Angostura bottles and other bitters with a flat cap. There is a plastic insert in the neck of the bottle with a small hole in the center that allows a drop to come out when held sideways or several drops when held upside down and given a sharp downward shake (a dash).

The third style is called a dasher bottle and is usually a bit fancier, often being molded with a pattern or cut crystal. The top is usually a stopper with a short metal tube that works similarly to the euro insert. It might look like a pourer spout but there is no air inlet so the liquid will come out slower.