LAMP Servers are the standard build for web servers. LAMP stands for Linux, Apache, MySQL, PHP. This gives you the operating system, http server, database server and PHP scripting language.
VPS , Virtual Private Servers, are simply virtual servers that are hosted by vendors such as Linode and Digital Ocean.
Virtual Hosts are used in Apache to allow a single web server to serve numerous websites. Basically www.FailedNormal.com point to /var/www/FailedNormal, and www.EliTheComputerGuy.com points to /var/www/EliTheComputerGuy.