Promise is both a verb and a noun, so you can use it in two ways.
"I promise that I will come back." -- This means that I am saying that I give you my word (promise) that I will come back. (VERB)
OR
"My promise to you is that I will come back." -- This means the same thing, but I use promise as the subject. (NOUN)
Using promising is a little different. It can be a form of the verb promise, or it can be used as an adjective.
"The most promising companies in the world had gathered for the conference." -- Here it means that the companies that showed promise/future success were all together at the conference. (ADJ)
You could also use it this way:
"The fact that she was promising good results was hopeful." -- This means that since she was giving her word/promise now, it was more comforting. (-ing VERB)
I hope it helped. If you have any questions, just ask :)