Martial arts is another complex skill introduced by the World Adventures expansion pack.  Superficially, martial arts is fairly simple - you learn how to fight, spar with other Sims, and eventually have the chops to beat up a mummy.  But there is a lot more to martial arts than just fighting - this may be the most complex skill available in The Sims 3.

Learning Martial Arts

Sims can pick up martial arts by getting trained by  a master, or by simply buying a training dummy and taking a crack at it.  (The training dummy is available for purchase in Shang Simla.)  Amazingly, Sims can teach themselves martial arts fairly well - please note that your real-life experience may vary!

Once you get good enough at martial arts, you can meditate.  Click on your active Sim and select "Mediate" from the option menu.  When meditating, your Sim will sit cross-legged, looking content.  And magically their martial arts skill will improve!

You can also start breaking boards by buying the board-breaking set in Shang Simla.  Start with foam boards and work your way clockwise around the menu.  As soon as you have broken five blocks, you're ready to move on to the next level.  If you push too far past your block-breaking abilities, you risk getting a broken hand moodlet!  

And finally, you can buy martial arts skill books from the booksellers in Shang Simla.  This would probably be the best option for a "Bookworm" or "Couch Potato" Sim to learn martial arts.  Otherwise I recommend learning by doing.  It boosts your athletic skill and lets you start working on some of the challenges while you learn.

Sparring

In order to get better at martial arts, at some point your Sim will need to spar with an opponent.  This can only be  done in Shang Simla, and only from certain locations.  (The Phoenix Martial Arts Academy is the only place I've been able to do it, but others say they can spar at the base camp.)

You can click on another Sim and ask them to spar.  The "spar" option will only appear if that other Sim has the martial arts skill.  

You can also invite over the next ranked opponent, just as you can with chess.  Simply click on your cell phone and choose Call Sim -> Challenge Next Ranked Martial Arts Opponent.  When the other Sim arrives, greet them, then click and choose "Challenge to a ranked match."

Once you have won 20 such matches, your Sim will be designated a Martial Arts master.  Woo hoo!

Side Benefits

Zeneport: when your Sim hits level 5, they can "zeneport" when they meditate.  Simply meditate until the game gives you the option to teleport somewhere.  Right-click on the destination with your mouse, and choose "zeneport here."  This has a lot of potential uses, including getting past traps while exploring tombs.

Breaking Space Rocks: after your Sim has broken 150 blocks, they will be able to break space rocks, and gems will come out!  Your best bet is to work with oak boards, since it's the number of boards you break and not their quality.  This can be quite a lucrative hobby for your Sim, and Mysterious Mr. Gnome can pop out of there, too!

Fighting Mummies: If you want to fight mummies in the tombs, then the martial arts skill is your only option.