Making a new Footprint Library in KiCad version 5.1.x
Footprint libraries are created from within the footprint editor. To create a new library you have two options. Either use the New Library entry of the File menu or of the right.click context menu in the tree view.
After pressing this button you will be presented with a file browser. The exact look of this depends on your operating system. Using this dialog you can select where you want your library to be stored. In addition there will be some form of text field that allows you to enter the name of the library.
By default the file browser will point you to the project directory. If you do not intent to make a project specific lib then it is highly suggested to select a different location. Much more details see above.
After this you will be presented with the option to add the library to either the local or global library table. Select whatever is appropriate for you usecase. See above for details.