There are lots of species, but only a handful get traded. Which ones you can buy largely depends on where you live: different species seem to be shipped to different parts of the world. Richard Mleczko goes into this at great length in my brackish water book. The Aqualog book also has some very nice photos of the different species you are likely to see.
For the most part, stick with the smaller Periophthalmus species. These are the easiest and most accommodating species. Some genera, notably Boleophthalmus, include species that have proven very difficult to keep alive in captivity. As
mlawson mentions, Periophthalmus barbarus is one you might want to avoid, not because it is delicate (it is incredibly hardy) but because it is big and has a very territorial and aggressive personality.
They are difficult to breed in aquaria because the males dig deep burrows and guard the eggs in them. As with many other gobies, the free-swimming fry will be small and likely require marine conditions and tiny live foods for the planktonic phase of their development. I'm sure breeding them could be done, but as far as I know, no-one has. If you want to breed gobies, there are several other species that don't have a planktonic stage and have proven much more amenable to captive breeding.
On the other hand, 'skippers are certainly not difficult to care for if you choose the right species. Many live in places with vile water quality, and things like salinity, filtration, and water chemistry are not especially important. Air humidity and temperature are far more significant, and essentially you need to create something similar to the conditions for keeping frogs or some other tropical amphibian.
Cheers, Neale
How many different breeds of Mudskippers are in the trade?
And can they be breed?