After you get the Legend Badge from Drayden or Iris, go to Route 7. You will notice that a huge rainstorm is in progress. Go to the house east of Celestial Tower, and an old woman will come out and invite you inside. After having some soup with the old lady and learning about Thundurus or Tornadus, you go outside, walk east, and end up face-to-face with Tornadus (in Pokémon Black) or Thundurus (in Pokémon White)! Tornadus/Thundurus will fly away to a random Route in Unova, and the storm will go with it. The storm keeps moving to random routes, so you will have to chase Tornadus/Thundurus around, weakening it until you can catch it.
In previous pokémon games, you could see roaming legendaries on the map. But in Pokémon Black and White, you can't see them on the map. Instead, the electronic bulletin boards in the route houses might mention "a very unusual storm" in a particular route. If you go to that route and there is a huge rainstorm, then that means that a random encounter in the grass might be an encounter with Tornadus/Thundurus.
Since you won't be able to chase Tornadus or Thundurus around because they don't show up on the map, just go into and out of a route house until the route has a huge rainstorm in it. Then you can go around in the grass to try to encounter Tornadus or Thundurus.
All roaming legendaries flee from the battle on their first turn. This means that you need to have a pokémon with decent speed in the first slot of your party. Whenever you damage the roaming legendary or inflict a status ailment on it, that injury/status ailment will still be in effect the next time you encounter the roaming legendary.
This means that you can gradually bring the roaming legendary's HP down, particularly if the first pokémon in your party uses False Swipe against the roaming legendary to make its HP exactly 1. In addition, you can inflict Sleep on the roaming legendary, and even in its sleep it will run away after the first turn of the battle, and will still be asleep the next time you encounter it. Therefore, it is pretty easy to get the roaming legendary into an easier-to-catch state.
The best pokéball to catch Tornadus or Thundurus is the Quick Ball. You can get these in Shopping Mall 9. Since the roaming legendary always runs away on the first turn anyway, using a Quick Ball on it will always give you the 4x modifier from being thrown on the first turn.
Otherwise, the next best is the Dusk Ball if the battle started after 8PM and before 3:59AM.
Otherwise, you'll have to make do with Ultra Balls, which have a far lower catch rate than the above two.
What can you do after catching Tornadus or Thundurus? Aside from training it, if you know someone who has the other version of the game and has caught the weather-based legendary from that game, you can ask that person to temporarily trade that legendary to you. After you complete the main game, take both legendaries to the Abundant Shrine in eastern Unova, and you will be able to catch Landorus, who is the third of the Kami Trio (with Tornadus and Thundurus being the first two.) Then you can temporarily trade your legendary to the person who did the previous temporary trade, and that person can get Landorus as well.