These days, when we use the computer for everything, it may be very difficult to find professionals who have beautiful and neat handwriting and whose work consists of using that skill. A century ago, however, such persons were easily found in all major cities. Their work consisted of writing on envelopes, invitation cards, certificates, diplomas, and other documents. Their neat and beautifully drawn handwriting was highly appreciated. What would such persons be called?
A calligrapher. There is the much lesser used word, "calligraphist."
1: a professional copyist or engrosser
2: one who practices the art of calligraphy
Calligrapher, as suggested by other answers, is most suitable for the majority of uses. A couple other words may apply, depending on the context:
A penman is essentially a synonym for a calligrapher, but may imply more of a skilled hobbyist than a professional. The term master penman has currency in calligraphic societies. The term gives rise to penmanship, a word describing (usually good) handwriting, especially everyday handwriting.
Hand-lettering used to refer to the custom drawing of signs and advertisements, although nowadays it has a more artsy connotation. So perhaps consider hand-letterer.
I believe the word you are looking for is "penmanship". Dictionary.com defines the word as such:
the art of handwriting; the use of the pen in writing.
a person's style or manner of handwriting: "clear penmanship; poor penmanship."
I hope this helps.