What it is the best way to make a chessboard for checkers using Kivy framework?
I have board.png, white.png, black.png, white_q.png, black_q.png files already. I wonder how to assign to each black tile on my board.png its own coordinate. Should I create  32 transparent widgets placed on black tiles of board.png or it is impossible? And what widget to use for 24 checkers? Any ideas or it is too complicated using Kivy and I should use tkinter?
There are many ways you could do this. It isn't complicated, it's very easy. The best way depends more on how you want to structure your app than anything else.

I wonder how to assign to each black tile on my board.png its own coordinate

Set the pos attribute of a widget to control its position, or better in this case use a layout that does what you want. For instance, adding your squares to a GridLayout with the right number of columns will have the right effect without you needing to worry more about positioning them.

Should I create 32 transparent widgets placed on black tiles of board.png or it is impossible?

I don't understand what you're asking here. You can make transparent widgets if you want but I don't know why you'd want to.

And what widget to use for 24 checkers?

The real question is, what do you want the widget to do? e.g. if you want it to display an image then inherit from Image.
Overall this answer is very generic because your question is very generic. I suggest that if you're stuck, try to ask a more specific question about a task you're struggling with, and give a code example showing where you are now.
Checkers board in kivy