"Handwriting" simply means what it says: writing by using your hand to hold a pen or pencil. "All students must use a word processor and print their essays. Handwritten essays are not acceptable. I don't want to try to read your handwriting."
"Penmanship" is the skill of good handwriting, or a noun describing how good or bad handwriting is. "A hundred years ago, everybody was taught penmanship in school." "I can't read his handwriting, his penmanship is terrible." Good penmanship is a practical skill, and simply means that the writing is clear and easy to read, not necessarily beautiful.
"Calligraphy" is handwriting specifically as an art form. Calligraphy reaches a high standard of beauty. Calligraphy is unusual in English-speaking countries. Most of us are not skilled enough to do it, most of us do not own the right kind of pen. "Sarah has taken up calligraphy as a hobby." "As a Christmas present, he gave her a framed copy of her favorite quotation, written in beautiful calligraphy." Calligraphy resembles medieval manuscripts, and may also involve arranging the words to form a pleasant shape.
In the United States, "handwriting" is often synonymous with "cursive handwriting," in which the pen is rarely lifted from the paper, there are no breaks separating the letters within a word, and the writing is a single connected line. Cursive is the opposite of "block printing," writing each letter separately as in print. There is a controversy going on today in U.S. schools, because with the advent of computers and keyboard, handwriting is less important than it used to be, and a few schools have decided not to even teach "cursive." Cursive is faster to write, but more difficult to read, and it is an additional skill to learn and some think time in school is better spent on other things.