You can read e-books on a huge variety of devices.
First, get your ePub or Mobi file on to the device you want to read it on (what type of book file do I need?). In most cases, you'll connect your device to the computer you downloaded the book to, using an USB cable, then move the book file to the correct place on your device.
E-reading devices vary, so here are some starting points for the most popular ones:
If you're using a dedicated e-reader, like a Kindle, Kobo, Nook, or Sony Reader, you'll able to read your book immediately. Otherwise, you should choose some suitable software.
Here is an extensive list of software to suit whatever device you may be using: