NuGet packages are extremely useful way to add libraries/components to your code base. This is similar to the traditional way of referencing DLLs except that you get notifications when there are updates.
You may find that if your code lives in separate Git repositories that if someone wants to use a project from one Git repo in another Git repo that they would need to download both. However, NuGet packages can be stored in a common location e.g. the NuGet Package Gallery which is available to all, or on a centralised private NuGet server. If you want to use a private NuGet server, you will need to add a package source in the NuGet package manager. You can do this by clicking the little cog next to "Package source".
Now for actually generating the NuGet package you can do this automatically on build, by opening a right clicking on a project and selecting properties. Then navigate to the "Package" tab and check "Generate NuGet package on build"