Hi Gabi,
jammr has the same requirements as computer recording. There are lots of tutorials and reviews available online that cover the options. I'll try to give a quick overview here.
If you would like to sing and play guitar then a USB sound interface like the Focusrite Scarlett Solo will provide good quality audio. This type of soundcard usually has one microphone and one guitar input.
To record singing you will need a microphone. Built-in microphones in laptops have poor quality audio, they are usually not suitable for recording good-sounding vocals. Instead a higher-quality microphone plugged into a USB sound interface through an XLR cable will sound good.
Acoustic guitars can be recorded through a microphone (even the same microphone as your vocals!). If you want to record vocals and guitar through the same microphone then it's important to check that the microphone you buy is designed for this, because some microphones are designed to focus on just a narrow angle and might have trouble picking up both vocals and the guitar.
Some acoustic guitars also have built-in pickups (electronics) so you can plug a guitar cable directly from the guitar into a soundcard without a microphone. If your guitar has a guitar jack then you can plug it into the soundcard and don't need to use a microphone.
Here is a video that shows you the options:
How to record guitarAlthough the video is aimed at recording songs on your computer instead of jamming, most of it applies. You won't need a DAW (like Ableton, Reaper, or Garageband) when using jammr.
To summarize:
1. Some web searches and YouTube videos will help you choose the right gear.
2. You can get good-quality recordings with a USB sound interface and a microphone connected with an XLR cable.