As far as I'm aware, the majority of plants require good drainage and don't like sitting in water. So what is the purpose of and what do I do with a pot that has no drainage holes in the bottom?
Like mentioned in the comments already, this is an outer pot where the inner pot goes in. So the inner pot is usually a plastic container with holes. When watering your plants, water drains through the holes and such an outer pot captures this water (just like a saucer would do as well). These outer pots are usually used indoor, so you can keep your windowsill clean. You ask (in comments), what to do with the surplus water after watering. Very simple, just wait like 20 minutes or so and remove the surplus water. Or try to dose the water so that there will not be any surplus water.
I use this system of inner and outer pots for almost all my indoor plants (see for example on my profile pic.). I think it looks nicer than containers put on saucers (but that's a matter of personal preference).