Looped cursive is the fully joined Continental school hand in which every ascender and descender is a loop, so the pen never leaves the paper inside a word.
Letterforms
Every tall letter loops above the midline and every descending letter loops below the baseline: l, h, b, k and f above, g, j, y, z and f below, with e, l and the entry strokes forming small loops of their own. The letters lean to the right and start with entry strokes. The result is a line with no gaps and no sharp corners, which reads as a single ribbon of ink when written well and as a tangle when written badly.
Joins and lifts
Every lowercase letter joins the next, and in most looped models the capitals join too, so a whole word is one movement. The loops are the joins: the upstroke of a loop is the connector from the previous letter, and its downstroke is the letter's own stem. Dots and crossbars are added after the word, and the pen otherwise lifts only between words.
Where it comes from
The school scripts of France and Germany in the nineteenth and twentieth centuries, descended from the copperplate hands of the eighteenth, kept their loops when the American models were simplifying theirs. French cursive and the German school scripts are the living examples; this site renders the model in a German school face. Looped cursive is also what most people picture when they think of old handwriting, and reading letters and records from before 1950 means reading it.
Practising it
The loop is the whole skill: practise l and e until the loop closes at the same height every time, then h, b and k, then the descenders. Keep the loops narrow; a wide loop tangles with its neighbour. The French cursive and German cursive pages describe the school versions, and the reading old cursive guide covers the loops that trip modern readers.