When installing new software you should use either the Ubuntu Software Center or
sudo apt-get install <applicationNameHere>. If you are downloading tars or zips, you are going to have a harder time finding/running them.
As for Java and Minecraft, run the following command from the Terminal:
Does it return your current version of Java? Or does it reply with
command not found? It should return output similar to this:
$ java -version
java version "1.7.0_45"
Java(TM) SE Runtime Environment (build 1.7.0_45-b18)
Java HotSpot(TM) 64-Bit Server VM (build 24.45-b08, mixed mode)
If it returns
command not found then you will need to follow the steps listed in this question to install Java: How do I install Java?
Once you have done that and are getting correct output from running a
java -version, then you can download Minecraft from here: https://s3.amazonaws.com/Minecraft.Download/launcher/Minecraft.jar
Minecraft (for Linux) comes delivered in a Java ARchive or JAR file. You can run this using Java. First, copy Minecraft.jar from your
~/Downloads directory to another location. I created a "minecraft" directory under my home directory for this purpose:
mv ../Downloads/Minecraft.jar .
To run Minecraft, (in the future) just change to that directory and invoke it with Java:
java -jar Minecraft.jar
Remember, Ubuntu (Linux) is case-sensitive, so make sure you don't alter the case of any of these commands. You can also do other things like give Minecraft (Java) more memory, or create a clickable desktop icon, but this should be enough to get you started.