You can upload your website via a FTP upload software like, FileZilla (free), LeapFTP (free), CuteFTP(not free) and there are many more. Using a FTP upload client is the easies and less complicated way of uploading your website to the server!
All you have to do is insert your hostname, ftp username and ftp password to the FTP client software and the connect to the server.
example:
Hostname = ftp.domainname.co.za
Username: yourFTPusername
Password: yourFTPpassword
Once you are connected to the server you will have to browse to your root directory, in cPanel it will be www or public_html and in plesk it will be httpdocs. Once you are in the root of your website space browse to your website and select all your files and click to upload the files to the server!
Once the files are uploaded to the server, you can visit your site and the files should be show up immediately. If it still shows the old site or the default webpage you can just press Ctrl + F5 at the same time and it will clear the cache in the browser and load the new contents!