Importance of fate maps
A good place to begin is with the experiments of Hans Speymann and Hilde Mangold, done in the 1920s. They transplanted a piece of the dorsal lip in a newt gastrual to the ventral side of another newt gastrula with different pigmentation. Because of this transplant, a "secondary" embryo formed, on what would have been the belly, in addition to the standard embryo expected. (Some picture show this to be rather like a second head.) Use of differently pigmented embryos allowed the researchers to note that the transplanted dorsal lip material developed into a second notochord. But the interesting facet is the newly developing notochord induced the development of a new neural tube in the adjacent cells which were otherwise destined to become "belly stuff".

My appologies for not finding a great reference with diagrams, my source is the 7th edition of Biology, Campbell and Reece. Here's a very thorough reference, with diagrams, if you're up for it.
https://app.shoreline.edu/kwennstrom/spemannmangold.pdf
This work contributed to a Noble Prize for Speymann. The bottom line here is that tissues receive information from adjacent tissues that influence their developmental fate. 
Later experiments showed that the inducing agent was chemical in nature