Requirements

Before we jump into action, it's good to have a closer look at Magento's requirements. What do you need to run it?

Simply put, all up-to-date requirements for Magento can be found here:

http://www.magentocommerce.com/system-requirements.

But maybe that's a bit overwhelming if you are just a beginner. So let's break this up into the most essential stuff:

Requirement	Notes
Operating system: Linux	Magento runs best on Linux, as offered by most hosting companies. Don't worry about your local test environment as that will run on Windows or Mac as well. But for your live store you should go in for a Linux solution because if you decide to run it on anything else other than Linux for a live store, it will not be supported.
Web server: Apache	Magento runs on Versions 1.3.x, 2.0.x, and 2.2.x of this very popular web server. As of Version 1.7 of Magento community and Version 1.12 of Magento Enterprise there's a new web server called Nginx that is compatible as well.
Programming language: PHP	Magento has been developed using PHP, a programming language which is very popular. Many major open source solutions such as WordPress and Joomla! for instance, have been built using PHP. Use Versions 5.2.13 - 5.3.15. Do not use PHP4 anymore, nor use PHP 5.4 yet!
PHP extensions	Magento requires a number of extensions, which should be available on top of PHP itself. You will need: PDO_MySQL, mcrypt, hash, simplexml, GD, DOM, Iconv, and Curl. Besides that you also need to have the possibility to switch off ''safe mode''. You do not have a clue about all of this? Don't worry. A host offering Magento services already takes care of this. And for your local environment there are only a few additional steps to take. We'll get there in a minute.
Database: MySQL	MySQL is the database, where Magento will store all data for your store. Use Version 4.1.20 or (and preferably) newer.

As you can see, even in a simplified format, there are quite some things that need to be taken care of. Magento hosting is not as simple as hosting for a small WordPress or Joomla! website, currently the most popular open source solutions to create a regular site. The requirements are higher and you just cannot expect to host your store for only a couple of dollars per month. If you do, your online store may still work, but it is likely that you'll run into some performance issues. Be careful with the cheapest hosting solutions. Although Magento may work, you'll be consuming too that need server resources soon. Go for a dedicated server or a managed VPS (Virtual Private Server), but definitely for a host that is advertising support of Magento.