For some, this is perhaps one of the best parts of the game. As a reminder, you can only catch Pokémon that you find in the wild. This may be Pokémon roaming around that you bump into or the more shy variety that hide in the shaking bushes. There are other ways to find wild Pokémon as well, such as fishing next to rippling water or by shaking berry trees; keep your eyes peeled!
Poké Balls at the Ready!
To catch a wild Pokémon, you must use a Poké Ball item during battle. You can do so by opening your Bag and navigating to the Poké Balls tab or, more easily, by pressing the X Button. Near the start of the game, you'll receive some Poké Balls for free, but eventually you'll need to buy your own from the Poké Mart clerk, found in every Pokémon Centre.
Once you use a Poké Ball, the wild Pokémon will become encapsulated by it. But it's not over just yet! Afterwards, the Poké Ball will shake up to three times. During this window, there's a chance the wild Pokémon will break out of the ball. If the Pokémon breaks free, you can try again. Otherwise, if the Pokémon remains in the ball, you've successfully caught it!
Improving the Odds
You'll have an easier time catching Pokémon if you consider the following three things.
The less HP the wild Pokémon has, the higher the chance of capture. However, you don't want to bring them to 0 HP, because they'll faint and disappear. To avoid that situation, it's generally a good idea to whittle down the wild Pokémon's HP with your weakest or next strongest moves. There are some moves like False Swipe that can't KO a Pokémon, which are perfect for this!
Additionally, you'll have a better chance if the wild Pokémon is affected by a status condition. Paralyse (triggered by Nuzzle or Thunder Wave etc.) is the easiest and most reliable method. However, it doesn't work on Electric-types. Otherwise, Sleep (triggered via Sleep Powder or Hypnosis etc.) provides a better chance, but the wild Pokémon may wake up.
Finally, it's important what Poké Ball you use. Great Balls and especially Ultra Balls offer a better catch rate overall. But there are special Poké Balls available via Watt Traders in the Wild Area or the right Poke Mart clerk in certain Pokémon Centres, which offer improved rates under specific conditions. Such as Net Balls, which excel at catching Water and Bug-types.
Upon catching a Pokémon, they'll be automatically added to your party of 6. If your party is full up, you'll have the choice of sending the newly caught Pokémon to your Boxes or one of your existing party members instead. Pokémon that you've caught will retain their HP, PP and status conditions from the last fight, so it's a good idea to heal them up before you use 'em.
Afterwards, you can manage your Boxes by using the PC in any Pokémon Centre or, once you've got the Pokémon Box Link, by pressing the L or R Button via the Pokémon menu. You'll be able to organise the Pokémon inside your Boxes, as well as move them to and from and your party. Unlike previously, Pokémon won't get restored when they're placed in a box, but they'll all be restored when you visit a Pokémon Centre.