Google Camera is the camera app used by Google on their phones (Pixel and Nexus). Compared to other brands, pictures taken with this app are usually more detailed and have better dynamic range, specially in hard conditions (low light, indoors, etc).
While Google Camera (GCam) itself can be installed on most Android phones, the features that makes it a good app don't work on non-Google phones. For example, it takes pictures, but it doesn't use Google's HDR+ tech which is what makes GCam good.
This mod or port was created to unlock these features on non-Google devices. Several developers and modders work to enable all functionality and fix bugs on other phones. Since Google made GCam to work only on their phones, a lot of work must be done for it to work on other devices.
What's the best version for you?
This is not the typical app you can find on the Play Store. For example, to install, you need to download the .apk file (file format used by Android apps) and then open it with a file manager. Don't worry, it's very easy to do.
You can start by trying the Suggested Versions. These are apks/versions known to work on many devices.
Another good way to find a good version for your phone is to check forum (eg: XDA) threads or groups for your phone. I keep a list of Useful Links that you can use. These threads or groups usually talk about the best versions, limitations, fixes, etc.
You can also check the Developers page. Under each username, you'll find the phone they own. If they own the same device as you, then probably their versions will work on your phone.
After finding a working version, keep an eye on future updates from the same modder as they are likely to be compatible with your device.
Why so many versions?
First of all, if the version you're using works fine, there's no need to try all new versions. If something is broken or if you want to try new features, then yes, keep testing new versions.
In order to improve photo quality, unlock new features, fix bugs on their phones, supporting new ones, etc, developers have to share their apks. Users are then able to test and provide feedback. Some of these versions work, some don't. Some are better than others, and that's why it's important to be part of a community where everyone gives feedback.
I upload everything to the download page because I can't test new versions on all phones. Sometimes a test version is a failure for a specific phone, but works very well on other phones.
Differences between developers
You'll see names like Arnova8G2, Urnyx05, BSG, etc, here. These are the nicknames of the people modding GCam. Some modders only support one phone, others support multiple devices. Some add new features, while others try to keep things simple.
A description about each one of them and links to their download page can be found on the developers page.