To chase something or someone means to follow and try to catch it or them.
When we use the word 'chase', we usually mean running after something with the intention of catching up. Imagine you're playing tag with friends, and one person is "it". The others have to run away from that person, who has to chase them to tag them out. It's a game, but chasing can also be more serious. For example, if someone loses their pet dog in the park, they might start chasing after it to try to catch up and bring it home safely.
To pursue.
To consume another beverage immediately after drinking hard liquor, typically something better tasting or less harsh such as soda or beer; to use a drink as a chaser
Example: I need something to chase this shot with.
To attempt to win by scoring the required number of runs in the final innings.
Example: Australia will be chasing 217 for victory on the final day.
To swing at a pitch outside of the strike zone, typically an outside pitch
To produce enough offense to cause the pitcher to be removed
Example: The rally chased the starter.
pursue
To groove; indent.
To place piping or wiring in a groove encased within a wall or floor, or in a hidden space encased by a wall.
Example: chase the pipe
To cut (the thread of a screw).
To decorate (metal) by engraving or embossing.