I'd like to give a little bit of attention to different ones individually. First, I thought I'd focus on what is known as "Magical Girl" manga.
Magical Girl stories are shoujo stories with the focus on a young girl heroine (usually a teen or pre-teen school girl) who possess supernatural, or "magical" abilities and fight the forces of evil. Usually they have a secret identity as an average school girl.
In most cases, there are special transformation sequences and specific abilities for specific goals. A few of the most common series are Cardcaptor Sakura, Sailor Moon and Princess Tutu.
Sailor Moon, one of the most popular and well known Magical Girl series.
While rarer, there are a few Magical Boy manga series that are being published such as D.N. Angel and Negima!
D.N. Angel, a popular Magical Boy series by Yukiru Sagisaki
Magical Girl series tend to follow four basic archetypes:
Cute Witch - generally considered the original Magical Girl archetype, the magic is usually mundane to the character. Sometimes she's otherworldly, other times she's a witch in training and learning to get a license to practice magic.
Idol Performer - the main character's secret identity is that of a popular pop singer, or idol. Oftentimes, they are other performance artists, such as a dancer or an actress and gained her ability either through magic or uses her natural artistic abilities to fight crime or solve mysteries. Fight moves tend to be sing offs or dance competitions.
Woman Warrior - Typically of the martial artist/superheroine archetype. This tends to have a broader appeal to general audiences. Oftentimes represents a period of growth throughout the series for the character. Has a number of mystical elements involved and the main characters is sometimes a reluctant fighter.
Phantom Thief - Are usually more fantastical in subject matter. The main character is more of an anti-hero and is considered a 'super criminal' who escapes without being caught or leaving any evidence behind. Tends to be more of a vigilante than a true criminal, even though the police feel they have an obligation to catch them. Uses trickery and has a strict moral code.
For more in depth information, please check out the Wikipedia page on the subject.