The 20th-century game of darts has dominated the use of this word, though there are some associated meanings, such as in dressmaking folds or paper darts. The original sense, in English from the 14th century, was exclusively military: a pointed weapon, either thrown by hand (as a spear or javelin) or shot from a bow.
The context in Shakespeare makes clear the military sense, but it's important to rid the mind of gaming associations when Martius talks of the air being filled "with swords advanced and darts" (Coriolanus, I.vi.61) or Jack Cade is described as having "his thighs (pierced) with darts" so that they resembled a porcupine (Henry VI Part 2, III.i.362).
It's much easier when the word is accompanied by adjectives which emphasise its hazardous meaning. Salisbury reports Prince Edward being surrounded with "crossbows and deadly wounding darts" (King Edward III, V.i.138) and Messala talks of Cassius as dead with "darts envenomed" (Julius Caesar, V.iii.76).