Sony also makes an Android app (not sure about iOS), which works very much like the Reader devices (minus a dictionary and annotations other than highlighting). The latest versions of the Sony Reader app let you sideload books, so you should be able to read anything on it that you were able to read on your Reader device whether it was purchased from Sony or not. While the Sony app doesn't support a built-in dictionary, it will let you lookup words and phrases on Wikipedia or Google as long as you're connected to the Internet.
I haven't spent much time reading from the Nook for Android app, so I can't comment on it other than it will let you sideload and has dictionary support. I know the Nook devices support
the same DRM as Sony (although B&N uses a proprietary DRM for books purchased from them), but I don't know if the app does.
The Kobo app for Android lets you sideload as well, but doesn't have dictionary support although it does let you annotate (highlight and add notes) as well as sharing passages on Facebook. I personally don't like reading much from the Kobo app because it doesn't paginate like the Sony (each chapter/section is paginated separately) and the annotation interface is a little awkward.
Kindle also has an app for Android, but as others have pointed out, it uses a completely different format from the others and you'd have to download your books in its format or convert (which isn't perfect, but works). I've not spent a lot of time reading on it either, but I'm pretty sure it supports sideloading and a dictionary.
If you decide to go with an iOS device, most, if not all of these apps are available there as well, but I can't speak for feature parity between the Android and iOS versions.