'Minor planet' is a general term for anything that orbits the sun, but is not a planet, comet, or man made object. Thus, minor planets include dwarf planets, asteroids, and centaurs... and occasionally, a misclassified comet.
While the term minor planet is still used, in 2006 the International Astronomical Union (IAU) officially reclassified the minor planets into dwarf planets and small Solar System bodies (SSSB). This cleared up the whole "what do we do with all these little planets!?" problem, but perhaps more importantly collected asteroids, centaurs, and comets into one grouping, the SSSBs. While this is more intuitive, we currently still use the traditional distinction dividing minor planets from comets when naming Solar System objects.
The naming conventions are somewhat complex, but very simply, comets are named year-of-discovery + half-month, but are often renamed after their discoverers if they are popular enough, while minor planets are named year-of-discovery + half-month, then given a leading number if confirmed, then given an unpredictable name to replace the date if people get around to it. (E.g. 1932 HA -> (1862) 1932 HA -> 1862 Apollo).