Although some anime fall short of their manga origin, many actually match the standard. An example would be the hilarity that is Soul Eater. In addition, those which are made as anime first (and then maybe made into manga) are also often good, Code Geass and Cowboy Bebop for exam...ple...
Now i probably ought to advertise them all... not my original intention, but okay!
Soul Eater is an ongoing manga series by Ookubo Atsushi, which is also being adapted into an anime. The story tells about a variety of characters, most notably Maka Albarn and Soul Eater, who attend Shinigami Buki Shokunin Senmon Gakkō, or Shibusen, who train as weapons and meisters in order to prevent the return of the being of insanity, the Kishin. With action and comedy all in one big stew, it truly is a marvelous series. The only downside is that the manga is only released once a month, while if you were hoping to watch the anime online, recently some of the episodes are being taken down, but don't let that stop you.
Code Geass is a completed anime series of two seasons created by Sunrise, directed by Gorō Taniguchi and written by Ichirō Ōkouchi, with original character designs by CLAMP, and has two manga spin-offs. In 2011, the Britanian empire, the last remaining superpower of the world, attacked and conquered the islands of Japan, using experimental robots known as Knightmare Frames. As a result of the conquest, Japan became Area 11, and it's citizens, now Elevens, were stripped of their rights and freedoms. Seven years later, the story focusses on Lelouch Lamperougue, a former prince of the Britanian empire, who obtains the Power of Kings: Geass, with the ability to bend the wills of others to his own. With his newfound power, he seeks to destroy his father's empire to avenge his mothers death, and to create a world safe for his crippled sister, Nunnally. Although some might pass this series off as another mecha, the robotic battles are nothing compared to the complex and gripping plot.
Finally, Cowboy Bebop. If you haven't heard of it, you're missing out. This anime series of 26 episodes, also from Sunrise, with two short manga series, were written by Keiko Nobumoto and directed by Shinichirō Watanabe. Taking place in the far future, where Earth is no more, but it's people populate the rest of the solar system, Cowboy Bebop tells the story of Spike Spiegel and Jet Black, two bounty hunters trying to make a living taking down the large number of criminals now thriving in the solar system. Not only is the story funny and action packed, the biggest part of it is the intricate and varied music that is brought to the table, each episode being themed to different music that complements the show.
I actually could've recommended many more, but in the end, i feel it's better to let people find new, exiting things for themselves, and not be completely led in one direction. Who knows? Maybe they'll find something and advertise it that I would take interest in...
Most of the mangas mentioned can also be found at
www.onemanga.com.
Code Geass and Cowboy Bebop can be found at good ol' YouTube (last I checked), though if you're having problems finding Code Geass, i can recommend
www.animeblogz.com, which also releases the new episodes of Soul Eater, though many of the old ones have been removed.