On each Mac there can be multiple Photos Libraries, but only one can be designated the System Library. The System Library is the Library that apps, like Messages or Screensaver or Safari, go to when they use a picture or save a picture in Photos. It is the only Library on a Mac that can synchronize with iCloud Photos.
To make one of your Libraries the System Library, open that Library (perhaps by double clicking on the Library) and go to Photos' Settings>General to see this:
Here, "Use as System Photos Library" is grayed out, since this Library already is one. If it's not grayed out, click on it to make this the System Library. Notice the Path to the Library above that to make sure this is the one you want and to see where it's kept. The default location is in the Pictures folder, like this one.
If you want to start from scratch, you can make a brand new empty Photos Library by using the Library Chooser. You get there by closing Photos and then option-clicking on the Photos Icon.
This shows all the Libraries that Photos knows about, where each one is located, and which one is the System Library. You can use this to Create and name a new Library. Then you can make it your System Library.
The System Library is not automatically connected to iCloud. You do that with Settings>iCloud.
If you have plenty of room on your Mac's hard drive, you should set "Download Originals" so that you can backup your pictures. If you're limited in space, then you can use "Optimize Storage," which holds only smaller versions of pictures on your Mac. The full sized files are kept at iCloud Photos, and Photos grabs them when they are needed.
When you make a System Library and connect it to iCloud, then all the pictures in your iCloud Photos will be copied to your System Library, and all the pictures in your System Library will be copied to your iCloud Photos. If you started with a new empty Library, then it will take several days for things to equalize. If you started with a Library that has pictures that weren't in the iCloud Library, then it will take longer. When you're not using the Mac, you should leave it plugged in, turned on, and with Photos not running in front. You don't what Photos to think that you are working on it.
The iCloud Library that is connected is the one for the Apple ID registered in the Library's User account. Be careful if your family has more than one Apple ID.
On a Mac, screenshots usually go to the Desktop folder. But on an iPad, screenshots go to Photos and, if iCloud is connected to the iPad, they are copied to iCloud Photos. If you haven't done so, you should go to iCloud.com, sign in, and look at iCloud Photos there. You should see everything that's on the iPad.
