"The raglan, from which these sleeves get their name, is a loose overcoat named after an English general."—The Complete Book of Sewing, 1949
Is it just me, or are sleeves fascinating? They have to fit such an odd assemblage of curves and angles: the arm is essentially a jointed cylinder that attaches to the torso by way of the shoulder, a body part that creates such an extreme curve that you have to shape fabric deftly around it by way of gathering, easing, and steaming. Raglan sleeves are just another way of maneuvering this territory. They're attached to a bodice by a seam that runs diagonally from the neckline to the underarm, rather than being set in at the shoulder. This pretty dress pattern shows the raglan lines very clearly.
Instead of a dart, you may also see a raglan sleeve in two pieces, with a seam that runs down the outside of the arm. In which case, the pattern pieces would look like this:
I especially love the raglan-sleeve swing coats of the 50s and 60s. See how this one has a seam on the upper side of the arm rather than a shaping dart?
Raglan sleeves are unusual in the way they're constructed in that everything is kept flat rather than applied in the round, as with a set-in sleeve. Usually, the underarm sleeve seam and the side seams of the garment are left open, until after the diagonal raglan lines have been stitched to the bodice. Then, the underarm sleeve seam and the bodice side seams are stitched in one continuous line.
Raglan sleeves can take all sorts of shapes and styles. On this 40s nightgown, they appear very soft and feminine.
On these coats, they're a little more structured. You can actually buy raglan-shaped shoulder pads if you want some extra shaping in a tailored garment.