Although Pokémon Go is a pretty complicated game, the basic game mechanics are simple. Trainers catch Pokémon and use them to battle other Pokémon at specific locations in the game known as gyms. Pokéstops and gyms are real world locations that trainers must visit to restock on supplies to continue on their journey.
Catching Pokémon
To catch a Pokémon, trainers use different types of Pokéballs, each with different catch rates. There are hundreds of different types of Pokémon in the game, and new ones are released every couple of months. One of the main goals of the game is to catch every Pokémon. Here is a video showing level 32 trainer Lorenzocv catching a Pokémon.
Gyms
Every trainer in Pokémon Go is on one of three teams, Mystic (blue), Valor (red), or Instinct (yellow). These teams use their Pokémon to fight for control of gyms, and leave a Pokémon to defend the gym.
Raid Battles
A raid battle occurs when a very strong Pokémon "takes over" a gym. Trainers come together to defeat this Raid Boss, and if they succeed, everyone gets a chance to try and catch it. The number of trainers needed to defeat a Raid Boss depends on the Pokémon and the level of the trainers.