Joomla has done most of the work here already. All you have to do is make sure your directories have the right permissions. Joomla even tells you that too.

In the horizontal menu click on "Installers" then "Components" which should give you something like this:

QUOTE:
NOTE! If you see anything that says "UnWritable" you need to check the permissions and make sure that directory has full write permissions, or chmod -R 777.

As you can see there are two ways to do it, one uploads a compressed file from your computer (RECOMMENDED) and the other grabs the files directly from your sever (files must be uploaded prior to the install process).

I use the first method, always. In fact I've never installed anything manually before, everything works using the normal method or it's not worth installing.

Click "Browse" to locage the component zip file on your PC and hit the "Upload File & Install" button.

If all was done correctly you will see a congradulations screen with a "Continue" link, click it, you're done.

If you have an error go back to the top and make sure the permissions are correct and that you are using a compressed component file.

QUOTE:
NOTE! Some components come with additional modules and mambots that are installed automatically during the install process. These must be "PUBLISHED" in order to use them.