Script and cursive are essentially the same thing. The word script is normally used for any kind of joined up writing. Cursive is a style of penmanship in which the successive letters are joined together. Caligraphy on the other hand is the art of fine/fancy handwriting; a script of a high aesthetic value produced by brush.
Flowing, cursive lettering is still seen today on wedding invitations and menus at fancy restaurants. While it looks extremely difficult to ink, it's actually made of two basic strokes.
Practice cursive letters A-Z with Education.com's cursive handwriting worksheets.
To learn how to write like the great Jane Austen, see Tutsplus.com's Mastering Calligraphy: How to Write in Cursive Script.
Of interest, find out what Beyoncé, Obama and Lady Gaga's handwriting say about them featured on The Guardian and see weird handwriting of very famous authors, compiled by BuzzFeed.