In English, we can “promise” or we can “make a promise.” These are very similar in meaning. “Make a promise,” however, has a little more emphasis, and sounds a little more serious.
When you marry someone, you make a promise to be there for them.
When you want to make a promise, you usually don’t say ‘I make a promise.’ You can just say ‘I promise’:
I promise, I won't laugh. Now show me the picture!
‘Make a promise’ is for talking about promises:
I made him a promise, and I intend to keep it.