Things that happened already are in the past, as in "Stop nagging me! I already cleaned my room."
If you did something already, you did it in the past, whether an hour ago or a year ago. A team trying for its second championship already won a championship. A kid who ate cake first has already eaten dessert. If you've never done it, then you can't say you already did it. This word often gets used in exasperated tones, as in "Take out the garbage! I already told you ten times!"