It depends on what kind of web space you have. If it's a VPS you will have a
public directory (or
httpdocs depending on the server software and configuration). You can upload the website to the directory that contains the
public directory and move the files from
public into the corresponding file on the server.
If you're on shared hosting, most of the time you have SFTP or FTP access to a folder that contains the public directory. You can upload your files here.
If the hosting only lets you upload into the public directory, you will need to keep everything (all classes, templates, etc) there. However, this configuration is very rare.