"Handwritten" is usually translated 手書き tegaki, for example
手書きの手紙 tegaki no tegami
a handwritten letter
(see ALC for many more examples).
Of course in handwriting, some characters are written less neat than the handwriting standard, but there is no standard "cursive" handwriting as taught in schools with the Latin alphabet with rules how to connect one letter to the next. (There is a cursive script, 草書 sōsho, but this belongs more to calligraphy than daily life handwriting.)
Some letters are written different in handwriting than in print — think
ɑ — notes on this are also contained in the official list of jōyō kanji (see pp. 7–10). (There are fonts which show this called 教科書体 kyōkashotai.)
However, to describe "normal" handwriting, I think 手書き is the most apt description. Finally, there are some common shorthand forms for kanji, which are referred to as 略字 ryakuji.