It is recommended you use a robust platform comprised of the Linux operating system, and either the Apache web-server or the NGINX web-server. Almost any server that supports PHP and MySQL will work. If your host doesn't support one of these platforms, and mod_rewrite, you will probably be better off switching to one of the many hosting providers that do offer those choices.
It is also essential that your host allows remote connections, for many of the WordPress features to work. If your host blocks outgoing HTTP connections, many parts of the WordPress will not function.
PHP 5.4 or greater is recommended.
Hosting is a commodity these days and with a little digging it's easy to find a host that supports the above. If you're looking for hosting suggestions, please try:
Self-hosting WordPress requires basic to moderate system administration skills. See Installing WordPress for setup information.