Serious doubts on Categorical embedding

I am still having many doubts about the working of categorical embedding.
In particular I have 2 points not clear:
1. Are 1-Hot variables converted to a lower dim vector?
2. What target are neural embedding trained on? Lots of explanations I have seen don't show any target, and I wonder how the back-prop is done.
3. It is not clear at all how the keras functional API works to build an embedding. Anyone is so helpful to explain step by step how to build one showing matrices dimension?
Thanks in advance.