|MadSci Network: Astronomy|
Thanks for an interesting question. There are a bunch of different ways celestial objects are named, and what follows is probably more detail than you want to know. Suffice it to say that, except for objects in our solar system, there's no standard procedure for naming things.
Some objects have "proper names" - like the north star Polaris, or the Ring Nebula, or the planets in our solar system. Sometimes these names are given by the discoverer and sometimes their origin is historical.
Some objects are named for the constellation in which they appear. These include most of the stars you can see with your naked eye: for example, alpha Centauri is the brightest star in the constellation Centaurus, and epsilon Indii is the fifth-brightest star in the constellation Indus. Some other things besides stars also get names this way; examples are the Andromeda galaxy and the Virgo (galaxy) cluster.
Another way things are names is by the catalog they appear in. For example, the star HD 225023 is star #225023 in the Henry Draper catalog. And the galaxy NGC 2403 is object #2403 in Deyer's New General Catalog (which isn't very new - it came out in 1888!) Many objects are in more than one catalog and so can have several names: for example, the Andromeda galaxy is also NGC 224 and Messier 31.
Solar system objects, such as asteroids and comets, have a well-defined naming system. Asteroids are numbered in the order of discovery, and also named by the discoverer, while comets are named for the discoverer(s) and year they were first seen. The Small Bodies Names Committee of the International Astronomical Union is responsible for these names.
The last way of naming objects is by their celestial coordinates: for example, the galaxy 16040+1552 is located at right ascension 16h04.0m and declination 15d52m. This method of giving names can be confusing, since there can be several objects which appear close together on the sky, and the "epoch" of the coordinates has to be specified (see this page for more info about astronomical coordinates).
As for getting information about an object based on its name, the only area I'm familiar with is galaxies. For that you can check out NED, the NASA Extragalactic Database. Similar things probably exist for stars and solar system objects - try looking in this list for a start. [Also check out SIMBAD.]
Try the links in the MadSci Library for more information on Astronomy.