I want to listen to books on my iPhone - what app(s) do I need to download?

I have taken up bicycle riding. Been listening to Pandora on my rides, but want to spend that listening time more constructively - listening to books.
What app or apps do I need (saw something about NOOK for listening to books?) And does the memory on my phone get used up with each book, or are they stored somewhere in the clouds?

You have several options. You can purchase and download books from Apple - they'll show up in the Audiobooks section of your Books app. You can join a service like Audible, which has its own app that you can use. Or you can download free audiobooks from Librivox.org and load them into the Librivox app or iTunes. There are also podcasts of audiobooks, which you can get with your Podcast app.

In all of these cases you usually have the option to download the books or stream them. Downloading is more reliable most of the time. I download one book, listen to it, then delete it and download the next. The file sizes are fairly small.

Here's what I would do… Do you have a library card from your local library? If not, get one. Then go online and sign up for an eMedia account. Your library should have tons of eBooks and audio books available. Different libraries use different software - could be Overdrive or Libby or you can just play on any media player. It will tell you and link you to the appropriate downloads.
Once you are set - it's free audio book checkouts from there on. They'll have a time limit - probably 2 or weeks and then they get sucked back from your device in some miraculous way.

Do not use that electronic trash, real printed old fashioned books are the best.

Yes, they're stored somewhere in the clouds. Later, they fall like rain down your window pane, so have an umbrella handy.