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Firstly let me begin by saying that WordPress and WordPress.com although related is not the same thing. WordPress (found at WordPress.org) is the software. WordPress.com is a company and is an internet (website) hosting company that uses a specific (in most cases a more limited) version of the software which has the same name. All you need to know is that in most cases the best solution is to use the WordPress software installed on your own server at a dependable hosting company.
WordPress is developed by the WordPress Foundation. Matt Mullenweg and Mike Little started work on WordPress more than ten years ago. WordPress is open source software released under the GPLv2 license.
WordPress is what is generally referred to as a CMS or content management system. This means that you have a place on your server where you as website administrator or author can securely sign in. in this administration dashboard you can upload media files (usually pictures) and write articles. this content then gets added to a database on your server.
You can also change setting that influence either the look or the function of a website from the administration dashboard. A CMS in effect splits the look and function from the content of a website to then combine those when displaying to a visitor. this means that you can at any stage change what a website looks like without changing the content that displays on the site. The other great thing is that once it is installed authors without any technical website or coding experience can add articles or other content to the website by themselves.
WordPress is a mature CMS and also the most used blogging system used on the web today. More than 60 million websites run WordPress.
Two features that make WordPress great for running a website are Themes and Plugins. Themes allow administrators to completely change the look and feel of a website. There are free themes available, premium themes can be bought or a custom theme can be developed specifically for a website.
Plugins are the second thing that makes WordPress really powerful. Independent developers can develop plugins for WordPress that are easy to install on a website and that extends the capabilities and function of a WordPress website. Plugins can also be free, premium or custom developed. Because WordPress is so widely used there are plugins available that accomplish almost any task that you might need done on your website.
In short if you are thinking of a new website or blog (yes WordPress is also a great choice for normal business websites and not only blogs) WordPress is in my opinion the best option. It is widely used, many people can give you support and help, is easy to extend and fast to implement.