Anyway, it occurs to me that the categories of shounen and shoujo anime, though useful for categorization purposes, are actually somewhat arbitrary, yet not necessarily clearly defined. it's one of those things that most longtime fans know almost instinctually, yet can't define concisely if you ask.
Since this is my website, and therefore my space to hold forth, rant, and otherwise get on the soapbox, I'll give you my definition (with examples!).
On the simplest level, shounen anime is marketed towards boys, and shoujo anime is marketed towards girls. Most anime is in fact geared towards children or teens, although there are of course a great many adult shows (read: pornographic) as well as a few of less explicit nature that are geared towards an older audience, often including shows that invoke nostalgia, such as, perhaps, Studio Ghibli's "Only Yesterday" or "Porco Rosso". I've also heard that the long-running anime TV series "Sazae-san" falls into this category. Still, these are only a few examples and they are in the minority.
That said, shounen anime is not only geared towards boys, it features boys as well, usually in a competitive environment, either a contest, a battle, or a quest. Well-known examples include Pokemon, Dragonball (and all the variants thereof), Megaman NT Warrior (a.k.a. Rockman.exe), Yu-gi-oh, and the various Gundam series (as well as most other giant robot series). In these series, boys or men take most of the prominent roles, while girls and women are mostly sidekicks, love interests, and friends. Though male/male friendship and comradeship is often a theme, shounen ai, or male/male romance is rare. Shoujo ai (female/female romance) is more common, though mostly in series that are aimed at a more teenaged audience, especially since shounen series are rarely romances. The art style also differs (in some cases) from shoujo series, typically being harsher and brighter, with thicker lines, darker shadows, and less in the way of scenery. The characters are also more likely to have spiky or oddly-colored hair.
Shoujo anime, of course, is more likely to feature girls. Romances are more common, both heterosexual and homosexual, though they not sexually explicit, and, in the case of homosexual romance, are likely more implied than anything. Shoujo anime based around contests are rare, if they exist at all (I myself have never seen one, that I can recall), and some shoujo anime (especially the magical girl genre) may feature girls (and sometimes boys) saving the world or protecting someone/something, without real battles. School dramas are also common, and, of course, pretty boys abound. Well-known examples include Sailor Moon, Cardcaptor Sakura, Revolutionary Girl Utena, and Inu-Yasha. The art styles used in shoujo anime tend to be softer, with more pastels, and water-color-like backgrounds.
These definitions aside, the most important factor in whether an anime is shounen, shoujo, or neither is the composition of its fanbase.