First you must access .minecraft, a hidden folder in My Documents/Application Data. Go into the saves folder to see all the worlds that are on your computer. Now you copy the worlds you want from your computer, put them on the other computer, and paste them into .minecraft's saves folder.
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In a sense, yes, but in another sense, no. You do not need to install Minecraft to play it on a computer, you can simply run it in your internet browser on Minecraft.net. However, if you want to run Minecraft without an internet connection, or without visiting Minecraft.net, you must install Minecraft on your computer. You can install Minecraft on multiple computers under the same account, but your saved worlds will not sync between computers.
You can bring a flash drive in, and put .minecraft and minecraft.jar in the school computer and thus run the .jar and play minecraft offline, on any worlds.
The saves folder is in a hidden folder in application data called .minecraft. The directory to the folder is: My documents/Application Data/.minecraft. There is a folder in .minecraft called saves which contains your saved worlds. If this folder does not exist, then perhaps you have not yet played any worlds on that computer or somebody has deleted the folder.
If you have purchased Minecraft you can download it onto any computer by going to the Minecraft website and logging in. However, multiple copies of the game are completelyseparate and saved worlds, mods andsettingswill not be transferred across.
Nothing will happen if you keep installing and uninstalling Minecraft. Minecraft does not track how many times you have installed it nor does track how many computers you have it installed on. The only drawback is if you uninstall Minecraft by deleting the .minecraft folder you will lose all of your saved worlds on that computer.