If you have Mac OS X 10.10 Yosemite, you already have a built-in recorder for your iPhone. To start, connect your iPhone to your Mac using a lightning cable and launch Quicktime Player. Click the File menu and choose "New Movie Recording." Click the arrow to the right of the record button and you'll see your phone as a choice. If you want to record audio as well, change the microphone to your phone as well. This will allow you to record whatever you're doing on your phone.
Not all apps will allow you to record the screen, such as Netflix. Additionally, switching from portrait to landscape will cause the recording to stop, so pick an orientation and stick with it.
You can also record just audio from your iPhone through Quicktime. For that, you'd go to Quicktime, click the File menu and choose "New Audio Recording." Then, switch the microphone to your phone. Quicktime by default has the volume all the way down, so you'll need to adjust it to hear playback.
This is a great way to show off apps or save something from your phone that isn't easily sharable. And these tips also work with an iPad.