An app is not an e-book. If your publisher brings out a Kindle edition of your book, that is not an app. If your publisher produces a pdf of your book, that is not an app (and it's probably not a publisher either - pdfs are for pirate copies and to give to printers, they are not for public consumption these days. How 1990s.)
App is short for application. If you have a sophisticated phone (a 'smart phone') like an iPhone or an Android phone (the Google phone software that competes with Apple), you can use apps on it. If you have an iPad you will use apps on it - you have to, it doesn't do anything else! From now on I'm going to call these iThings and mean all of them. You might have apps to help you find your way when you get lost, apps to play games, apps to make stupid noises... I even have an app to turn my iPad into a spirit level because I couldn't find the real spirit level when I was putting up a towel rail. It's not much use day to day, except perhaps as a sobreity test.
Now, you can get book apps and you can get apps to read books. These are not the same thing. An example of an app to read books is the Kindle software which you can install on your iThing . You can then get books from the Kindle store on Amazon and they will be electronically schlurped into the Kindle app so that you can read them on your iThing. These books look rather like a printed book, and are effectively a printed book that is just read on a screen. There are a few things you can't do with a printed book, such as change the size of the print, and a lot of things you can't do with a Kindle book that you can do with a real book, such as squash wasps. But the model is 'let's make a book you can read on an iThing' and that's all. Nothing extra. Oh, and you can install the Kindle reader on your computer if you don't have an iThing.
Now we get to the point. An app that is a book is free-standing. You don't open it in Kindle or any other bit of software as it is itself a bit of software. So you can think of a Kindle book as being like a Word document and you need Word (or in this case the Kindle reader) to open it. But a proper book app runs on its own. It's like playing Minesweeper or Halo III or Solitaire - it doesn't need any other software. Some other analogies: a Kindle book is like a virus (has to exist in a living cell); an app is like a bacterium (can survive on its own). A Kindle book is like an airline passenger (not going anywhere without a plane); an app is like a pedestrian (self-propelling). A Kindle book is a tapeworm; an app is an earthworm. Got it?
While a Kindle book emulates a real book, an app is something altogether different. It can and should provide a lot more than just some text. There may be pictures, video, sound, animation, and interactivity of various types. ('Interactivty' means you can do things, like drag stuff around the screen or tap on it to make things happen.) The books that work best as apps are generally picture books and illustrated non-fiction. The flashy add-ons help to engage a small child and draw them in to explore the book and develop their reading, or to add extra information of types that can't be put across on the printed page. You want to see how a spider scuttles, hear the noise of the space shuttle taking off or watch a schematic animation of the movement of a piston? An app can do that, but the printed page can't. A lift the flap book is delightful - but if the monster scuttles from under the stone and hides somewhere else when you 'lift' the stone, the game can go on. In marketing speak, apps add to the reading experience. This can be valuable - or it can be insulting and a pain, but no-one made you buy the app.
Apps of novels are a different matter. The joy of reading a novel is using your own imagination to create the parts the author has left out. Yes, it might be useful to hear the piece of music described, or to see the historic monument visited. But I don't want to see someone else's idea of what the characters look like or to hear their voices (ie an actor's voice) or see the inside of the rooms or the path they walk down. Those are my contribution to the novel, they occupy the space the writer has left for my creativity and I don't want them invaded by someone else's ideas. It's like seeing a film of a novel - it's very hard to get the new interpretation out of your head. So I won't be buying apps of any novels in the near future. But that's a personal choice.
As a writer, why should you care about apps? Well, you need to be careful which rights you sell and what they mean. Most publishers want electronic rights to your books when you sign a contract these days. These can be limited in various ways - eg the publisher has the rights for two years and if publisher doesn't exploit them they revert to you. Electronic rights do not automatically cover apps rights, which should be a subsidiary right (like TV, merchandising and film rights). You should endeavour to get apps rights specifically excluded from the deal, as most publishers are not doing anything at all with these rights in most books. You don't want, really, to tie your rights to a heavy stone and chuck them in the canal.
If your book is to come out as an app (and you didn't write it with that in mind), there will be a lot of extra work for you to do. Or there should be. There should be consultation on images and sound, a request for extra material from you, and so on. You will need to negotiate payment for this, and the royalty you expect from the app. Here you are in uncharted territory, or at least the lawless wild west of publishing. Many writers feel they should get a higher royalty for e-books than paper books (in reality this depends on the format, and whether the book already exists in paper format, in which case conversion is cheap). Apps are different. A good app costs more to produce than a paper book. But a bad app is cheap to produce. If all the app does is read out the story and show some static pictures or very simple animations, it doesn't cost any more to produce than a full-colour paper book. If it has lots of interactivity, animation, sound, multi-threading (ie you can use the 'pages' in different orders) and other bells and whistles, it will cost a lot to produce. It's open season - negotiate strenuously.
But be aware that if you get an all-singing, all-dancing app, the publisher will have a lot of costs to recoup. Another problem for publishers is that apps are platform-specific. This means that if an app has been made for Apple i-Things it won't run on Android i-Things. An app made for Android iThings won't run on Apple iThings and it won't even run on most Android iThings as there is little consistency between the devices. You could think of it as being like different language versions - if your book is printed in English, it's not going to be much use to someone who only speaks Arabic. But it's a bit more complex and the app has to be rebuilt to run on different platforms, then tested all over again... it's a costly process.
This means that for all the shouting about apps, they're not actually making anyone very much money yet. Publishers think they have to do it, but most publishers have little experience of software development and don't have a clue just how many worms are in the can they're opening. There will be casualties... One problem, especially for small publishers who are having to depend on freelance app designers, is that software people don't live by the same rules as publishers. They are hard to manage. They are likely to take a dislike to a publisher or project or become bored and dump them. Which they can - they finish the current app and go off to Africa for six months because it's more fun to see elephants than to animate imaginary elephants. It's not going to be an easy ride.
To see some apps in action, if you don't have an iThing, look on YouTube for reviews and demos. Here's a (rather frantic) demo of the famous Alice app, which was the first really to exploit the (Apple) platform but is actually designed to show off what can be done rather than add anything for target-age readers. And here is a review of Dr Seuss's Cat in the Hat. This is an underspecified app - it does pretty much nothing except read the text aloud and annotate the pictures with some useful words. And here is one that does as it really should and makes creative use of the technology in a way appropriate and enhancing for the readers: Oliver Jeffers' The Heart and the Bottle. The picture at the top of the page is from Feridun Oral's Red Apple published by Winged Chariot. Their apps offer the text in a range of languages - that's a good (if obvious) use of the technology.