XAMPP is a package that installs Apache, MariaDB, PHP and Perl onto Windows, Linux or Mac. It is a very simple way to get a web development environment up and running with little knowledge.
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.