What is the software that the Magento server needs to run?
The full list of software is listed in System Requirements but here are the essentials:
Web server to display web pages. Currently, we’ve documented Apache and plan to document other web servers in the near future.
Database management system for long-term data storage and retrieval. We support MySQL.
PHP, an acronym for PHP: Hypertext Preprocessor, an open source scripting language suited for web development because it can be embedded in HTML (the language of the web).
If you use a hosting provider, they might already have installed this software. Contact them if you're not sure.