Handsome is generally a good looking man, while Beautiful is generally a good looking woman, though of course some of these gender rules break down as they are not very strict. Beautiful is also used to describe things as "good looking".
Nice, is more complicated as it has a range of meanings
Normally it is a very fuzzy pleasant description of something or someone. So "she is nice" is just positive, it can mean she is a pleasant person or it can be she is slightly pretty.
Nice can also been "precise" so a nice project could be just a a good project or it could be a very clean/tidy project.
Because of this uncertainty main native speakers are brought up being told "don't use 'nice' as it doesn't really mean anything"
Examples, he is a handsome man, she is beautiful. What you did for your friends was nice.