Oil, watercolor and gouache are types of fine art paint, such as for use on paper or canvas. None of those are really what you want for painting models.
The old-school model paints such as Testors are enamels. While they can be thinned or cleaned with thinner/turpentine like artist's oil paints, they're not really the same thing. While they work just fine on models or miniatures, most miniature painters abandoned them long ago because they're a pain to work with (they're toxic, require paint thinner to clean up and take a while to dry).
Almost all miniature paints are acrylics. This includes all brands you've probably heard of, such as Games Workshop (aka Citadel), Vallejo, P3, Army Painter, Coat d'Arms, etc. So-called "craft paint" like you might find at a department store is also acrylic. Acrylics are so popular because they're easy to use - they thin down and clean up with water, will dry in 10-15 minutes and are usually non-toxic. Artists also work with acrylic paint for much the same reasons. Artist acrylics are usually fairly thick and sold in tubes, while craft paint and miniature paints usually come in a pot or dropper bottle.
The difference between craft store acrylic paint, artist acrylics and miniature paint come down to the different formulas of acrylic binding agent they use, and also how the pigments in the paint are ground and mixed. Miniature paint has acrylic base that is formulated to be as durable as possible, to help the paint stand up to being handled when you pick up and play with your figures. They also tend to have finely-ground pigments, so the paint will stay smooth if you thin it down a lot. Artist acrylics will usually have fine pigments, but the acrylic that holds the paint together isn't formulated to protect against handling (how many times do you pick up a painting compared to a miniature?). Craft paints can be a mixed bag - some companies have perfectly fine acrylic and pigments, while others will try to cheap out to cut costs.
All these types of acrylic paints can be used to paint miniatures, you just need to keep in mind the differences. You can certainly do most painting with craft paint and use miniature brands for more specialized stuff like metallics and washes, but you'll probably need to thin down the paint a little more since most craft paints are thicker out of the bottle. Artist acrylics can work too (I've seen it done) but keep in mind that you may need to do more afterwards to protect the paint from handling (such as extra clearcoats over top). Most people prefer miniature paints because they're already formulated for the job and therefore more convenient.
Citadel paints are popular because they're an easy match for lots of existing color schemes in the game, and their base/layer/shade system makes it easy to know what paint to use for what technique. Vallejo is another popular option because they have a huge range of paints available and are less expensive than Citadel. The good news is you can easily mix and match between different acrylic paint ranges - I have some models that have Vallejo, Army painter, P3 and Citadel paints on them just to get the color or effect I want. Which black I use is usually dependent on which is closest to me on the desk at the time
