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.