The word handsome is easily understood and commonly used; It means “good-looking,” with high-class and masculine associations—i.e. typically a man who is dressed and groomed like a “gentleman” (neat, clean, wearing non-casual clothing).
Handsome can also be used to describe a woman, sometimes, but still carries a masculine connotation, versus “pretty,” which has feminine connotations. A handsome woman has the same characteristics as a handsome man, including a little bit of masculinity.
Hansom, on the other hand, refers to a kind of horse-drawn vehicle—a low, two-wheeled, covered carriage, in which the driver sits up high on a seat behind the passengers. It is not a commonly used word, and not even known by many English speakers. However, as hansoms have become more common in American cities recently—for taking tourists on rides—perhaps we’ll see an uptick in its usage.
Note: hansom sometimes appears as a misspelling of handsome, even in print, so make sure you’re using your handy spellchecker.