Meteors appear at random times in game.
When a message flashes telling you that a meteor has flown by, get out your explorer spy glass(that you can buy in game from various npc’s).
This spyglass will tell you the direction of the meteor. Once you get to the location, you can begin mining for the meteor.
Meteor ore is used in the same way that regular ores are, you smelt them into ingots and then craft into whatever item you wish to make.