PDF uses Adobe Reader and is the most versatile format, but it will need support software to use this format on an e-reader. Recommended for reading on your computer.
Most ebook readers that aren't the Kindle use epub formatting, including the Nook.
.mobi is specific to the Amazon Kindle.
However! Amazon doesn't allow users to download files to their library if the book doesn't come from the Amazon bookstore. To get books from third party sources (like us) you'll need to download the file on your computer and transfer it to your kindle manually.