Download the latest OpenCart version from here.
Extract the archive somewhere.
Upload via FTP, the files under the “upload” folder.
Go to the CPanel of your account and create a SQL database, a SQL username/password (if you don’t already have one). Assign the new SQL database to a username.
Write the url of your domain (the new store), in the browser.
You should see the installation process.
Click next, you should only be required to write the SQL database, the SQL username and the password for that username.
That is pretty much it.
Remember to create a robots.txt file and write:
User-agent: *
Disallow: /
so Google and other search engines will not index your website for now. At least the default store that comes with OpenCart which has generic products.
The admin page is yourstore.com/admin. The username and password are “admin” (default) and the password was set by you during the installation process.
If you need more help, leave in the comments.