It's a piece of software that you load onto a web server which allows you to create a website without having to write the individual pages in HTML and CSS. It's especially useful in situations where the people who will be adding or updating content on the site don't have HTML/CSS skills.
A Worpress site can be run on any server that meets the requirements. This can be on your own web space that you rent from a hosting provider (in which case you have to install and maintain Wordpress yourself). There are also sites (such as wordpress.com) which provide hosted Wordpress accounts, where you sign up and get access to a pre-installed version. This is less flexible than hosting it yourself, but requires much less technical knowledge.
The look and layout of a Wordpress site can be altered by installing different themes (there is a huge marketplace for 3rd-party themes to suit all tastes/budgets), and themes can be modified if you know HTML, CSS and a little PHP. Other aspects of the program can be changed by installing plugins (again, there's a huge marketplace for these too, both free and paid).
No, and if you manage your own installation of Wordpress then no 3rd-party has anything to do with your system or the users you add to it.
I hope that helps to answer your questions. If there's anything else you want to know please ask.