I've been thinking about getting an e-reader for myself, and have been doing some research.
I've learned that there are two major formats, Kindle and epub. Kindle format is proprietary and belongs to Amazon.com. You can only read Kindle format on a Kindle reader. epub is a general domain-type format that anyone can use, or publish in, and there appear to be more free classics and old books, including Catholic ones, available in epub than in Kindle, although there are a good number available in Kindle format. Nook is a popular ePub reader, but there are various others. Kindle readers can't read ePub books, and vice-versa. epub readers seem to cost slightly less than Kindles, but it depends on where you find them and if they're on sale, etc. The price difference usually isn't very significant, from what I've seen.
The more expensive e-readers such as the Kindle Fire are more like tablets, and let you download and watch movies in addition to books. They also usually have a simple browser. A lot of them have 4G now, which means you can access the internet pretty much anywhere, and not just in a Wi-Fi hot spot. Also, the color ones are harder on the eyes than the black and white ones, which use "e-ink" and replicate the experience of reading on paper more closely than the color e-readers do. Nook offers a black and white epub e-reader with a backlit screen you can turn on for reading in bed in the dark without waking up your spouse. It's about $100, and is the one I'm thinking of getting. It doesn't make you watch ads, and won't let you watch movies or browse the internet.