The first thing you have to understand about fountain pens is that the ink basically has the viscosity of water, it isn't like other "ink" in ballpoint pens or gel pens. A fountain pen has to essentially function as a controlled leak to write... while not leaking.
When ballpoints came into the picture and steamrolled fountain pens (as the utilitarian writing tool) the methods of creating a vessel to hold ink inside a fountain pen without creating a mess/leaking were pretty primitive/unreliable by todays standards. A common solution was to just fill the hollow body of the pen entirely up with ink and then put silicon grease on the threads where the nib screws in (it could leak out). The easiest way to fill a narrow, light cylinder with ink you REALLY dont want to spill is with an eyedropper type device, hence the name eyedropper.
People still do this with fountain pens, apparently fountain pens are decently popular in india and a lot of indian fountain pens are eyedropper pens.
Most fountain pens these days are what are called "cartridge converter" pens. The name is weird, but the original innovation over crude rubber sacs that you would squeeze to suck up ink (itself an improvement over eyedropper style filling) was to make plastic cartridges that could be filled with ink, sealed with wax and then inserted into the pen.
Another big innovation was piston filler fountain pens that have a piston on the inside of the pen body that can be moved in or out by rotating a knob at the end of the pen. Not only is this an improvement because you can stick the pen directly into the ink and just suck it up through the nib by retracting the piston, ink can be manually advanced out into the nib/feed if the pen was writing dry, and in the opposite sense there is always a bit of suction keeping the ink in that you can adjust. A fountain pen's "feed" is basically a big capillary force engine, and it is nice to have a counterforce with the piston that can be adjusted to either aid or inhibit it.
So then someone took the whole piston filler idea and minituarized it so it could slot into pens designed for cartridges, hence the name "cartridge converter" pens because these self contained piston fillers were called converters.
Eyedroppering pens is something people do for fun still, its an ok way to fill a pen if you dont care about the pen heating up as you hold it, creating a pressure differential and "burping" ink out onto the paper occasionally.... its actually far safer to keep an eyedropper pen mostly full so that there is less of bubble of air to heat up and cause this.