Make sure that you fullfill the minimum requirements to run Coppermine on your webserver.
You can (and should) rename the coppermine folder, but not the files or folders within.
You need an application that can unpack an archive like 7-Zip if your OS doesn't support unpacking out of the box (Windows XP or better comes with support for zip files; most Desktop Linux distributions come with corresponding clients like File Roller as well). In some Packing/Unpacking Applications like WinZip there is a toggle that allows the end user to specify if the folder structure within the archive should be kept intact or if all files contained in the archive should be unpacked into one flat folder without sub-folders. For Coppermine, the option that preserves the folder structure must be enabled.
A good place to unpack your copy of the Coppermine package to is the Desktop or another temporary folder. In this explanation, we'll assume that you have unpacked the archive on the Desktop.
After unpacking, you should have a folder on your desktop named cpg15x. You are welcome (and encouraged) to change the name of this folder to something that makes more sense for usage on the internet, but make sure to use a web-safe name: don't use special characters except the dash (-) and the underscore (_), don't use spaces, don't use accented characters, Umlauts or any other special characters. Avoid capitalization if possible. In fact, you should use latin alphanumerals only (a to z and 0 to 9). This recommendation does not only apply for the Coppermine folder, but any folder or file you upload to a web-server.
Use your FTP client application to upload the folder you unpacked in the first step to your webserver. Do not use browser add-ons or any other crutch to upload the files, but use a real FTP app. Do not trust uploaders that often come with WYSIWYG-editors like MS Frontpage or Dreamweaver, as those built-in uploaders tend to "beautify" the code files, crippling them that way. Those editors and the tools they come with are OK for other purposes, but they often cause issues when used with pre-made scripts like Coppermine, that's why the Coppermine dev team strongly recommends not to use such WYSIWYG-editors at all. If you don't have an FTP app yet, take a look at the list of tools recommended by the Coppermine developers.
To use your FTP application, you will of course need to know the FTP details for your webspace. If you're not sure about those details, ask your webhost for support.
Make sure to use the correct FTP mode when uploading.
If you plan to make Coppermine your primary application that shows on your site's start page, you don't necessarilly have to put the Coppermine files into a separate folder: you can savely upload the content of the Coppermine package into the webroot of your server,
After performing some basic checks, the installer creates the needed database tables for you and fills them with default values. It creates the file include/config.inc.php within the coppermine folder on your server that stores the database details you entered during install. If you should change your mysql database details later (i.e. if you change the password of your mysql user account or if you migrate your gallery to another server), you will need to edit include/config.inc.php manually to reflect the changes. The file include/config.inc.php also keeps the install script from being run twice: if the installer is run, a check is performed wether the config file exists - if yes, the installer will stop and redirect the user to the index page.
There are some additional pieces of information that are related to installing:
When installing Coppermine, you might run into issues not directly related to Coppermine, but to webserver setup. Therefore, this section of the docs can only give a rough overview about the webserver setup issue. We can only give recommendations, but it's beyond the scope of the coppermine support board to actually advise how to configure a webserver in the first place. This being said: if you have issues related to webserver setup, ask your webhost for support instead of asking your question on the coppermine support board.
In older versions of PHP, the default setting for register_globals used to be "on", which resulted in many PHP coders coming up with sloppy code in terms of security (simply because they were not aware of the dangers that lay in such sloppy coding). As a result, there used to be many scripts that relied on the server variable "register_globals" to be turned on. However, as many malevolent attacks against scripts are being carried out that rely on register_globals to be turned on, it's advisable to turn that feature off in the config of your webserver's PHP settings. In recent versions of PHP, the default setting is "off". However, your webhost may still have enabled register_globals to allow you to run outdated scripts that rely on register_globals.