If using an iPad goto the Apple App store and find the Kindle app. Donwload and install then log into your Amazon.com account. If you do not have an Amazon.com account you will need to goto Amazon.com and create one. One you install the Kindle App on your iPad and log in Amazon will "register" your device and you can then purchase Kindle eBooks and sync them to your iPad.
Use the same method for any Android device. Goto the Google Store and find the Kindle App if not pre-installed. Download it and install it. Log into your Amazon.com account and sync the eBooks you bought, like above.