A lot of the naming conventions were originally "because they remind us of things we already called this", or simply "tradition". How we name things has slowly but surely adjusted with time as more objects were found, and a more robust classification system was needed.
Imagine it like having bins to sort your toys into. If you have a small number of balls, perhaps you'd stick them all into a single bin called "balls". But as you get more and more balls, you find they no longer fit into that bin. Looking at them, you notice they are all either quite small, or quite large. So you make two bins: small balls and large balls. Or maybe you notice they are all either very soft (stuffed or Nerf versions, perhaps), or very hard (baseballs, for example). So you could have a "soft balls" and "hard balls" bins. Keep collecting ever more balls, and you may need three, four, etc. boxes to fit them all in. And each time you will naturally try to order them so that each box contains balls that are similar to each other: these are all soft and green, these are all hard and white, these are all big and bouncy, etc. This applies equally well to all sorts of collections: baseball cards may start all together, but then get sorted into teams and years or even individual players as a collection gets very large, and so on.
This is basically what's happened as we've classified and reclassified objects we've found in our solar system. We started with a small number of objects, but as we found more and more of them it became too unwieldy to stick them all together, so we started to separate them more and more.
Ceres was labeled a planet because it seemed like a tiny version of, but otherwise very similar object to, existing planets. Then we eventually discovered there are a lot of things in the same general orbital region as Ceres, and if those were to also be planets then we'd soon have a gigantic list of planets. Since all these objects seemed pretty similar, and came from the same general region, they were given their own class of objects: asteroids.
Pluto was labeled a planet due in part to some initial errors in the data that suggested it was larger than it actually is. Despite a growing number of oddities about it that made it look increasingly un-planet-like, tradition left it as a "planet"; it wasn't particularly problematic for this one little odd-ball to be lumped with the others. Finding a host of new, similar, and sometimes—in the case of Eris, at least—bigger objects forced a reconsideration.
Incidentally, there are also comets, which are not the same as asteroids. Asteroids are rocky objects in the inner regions of the solar system. Comets are icy objects from the outskirts of the solar system: the Kuiper belt and Oort cloud, in particular. They are vaguely similar to each other: generally very small, lumpy, potato-like objects, but we noticed that asteroids were mostly rocks and metals between Jupiter and Mars, and comets were mostly ices beyond Neptune. So we separated them into their own groups.
Pluto is more comet than asteroid, and likely originated from either the Kuiper belt or Oort cloud. In this case, it likely got knocked into a closer orbit via gravitational interactions with the gas giants, or possibly a star that got close enough to the Oort cloud.