TYPO3 is a Web Content Management Framework, that is: a tool to develop and maintain web applications. It is very highly customizable, and has an modular structure that allow for adding and modifying behaviour of the system easily.
TYPO3 is an open-source project under the GPL license. This means that the intellectual ownership of the code belongs to the developer who wrote it, but anybody is allowed to use and modify it as they please within the terms of the license.
TYPO3 was created and for years primarily developed by Kasper Skårhøj. Now, the core team has taken over the responsibility for the quality of TYPO3. No single company is responsible for the development of TYPO3, but the non-profit and non-commercial TYPO3 Association, now organizes and supports the development. The project is highly dependent on the contribution of the users of the software to keep it's high level.
Yep, download your own copy.
You could either:
Pay somebody to develop a TYPO3 website for you. There's no "TYPO3 company" that you can get in contact with, but there are lots of web agencies and freelance developers all over the world.
or
Get started yourself, this requires: Some technical knowledge of web development and also time, motivation and patience to learn about web technology in general and TYPO3 specificially, and possibly a lot of coffee. Start here.