Don't go into the tall grass! Oh, you have Pokémon; you should be OK.
In all seriousness though, it's a dangerous world out there, and you need to know how to prepare yourself by catching Pokémon! First, you will need Poké Balls.
A detailed guide on Poké Ball making can be found here.
After you have the necessary equipment, find the Pokémon that you want. Keep in mind that Pokémon spawn in different biomes and times (split into day, dusk, and night) and that some are rarer than others. Some Pokémon are not even found in the wild at all, most of which are evolved Pokémon. A few others, namely Legendary Pokémon; Fossil Pokémon; and Porygon, are obtained through special methods. Also note that you can't catch boss Pokémon or Pokémon owned by other Trainers.
Once you have found your target Pokémon, send out your Pokémon and battle it. Although it is possible to just throw a Poké Ball and catch it right now, weakening the Pokémon will increase the chances of capture. Use moves that you know will not defeat the opposing Pokémon, as they can't be captured if they faint. If you are able to inflict a status conditions such as paralysis and sleep, your chance of success will increase further. Different types of Poké Balls work better in different situations as well. Keep in mind that the difficulty of catching different species of Pokémon varies according to their catch rates. Once they are weakened, go to your bag in the battle menu and select a Poké Ball, or any ball of your choice. If it is a successful capture, the Poké Ball will stop shaking and a message will appear saying that the Pokémon was captured. The Pokémon will appear in your GUI on the left, or be deposited in the PC in the event that your party is full.
It is also possible to just throw a Poké Ball outside of battle by using the item. This requires you to aim at the target Pokémon, and if you miss, there is a chance that the Poké Ball will break into its components parts, necessitating the reassembly of the Poké Ball.