A stole worn straight down from both shoulders means a bishop or a priest. A stole worn diagonally, over the left shoulder and fastened at the right hip, means a deacon. In the Eastern churches the deacon's equivalent, the orarion, is worn over the left shoulder with the ends hanging free, and the subdeacon crosses it. That single difference of angle is the most economical statement of order in Christian dress: three ranks, one garment, distinguished by how it hangs.
Where the garment came from is genuinely unresolved. One account derives it from the orarium, a cloth or napkin carried by Roman officials and used for signalling; another from a scarf of office worn by imperial functionaries, which would make it an insignia borrowed from the civil service rather than a liturgical invention. The evidence does not settle it, and the vesting prayers that accumulated around it, which speak of the yoke of Christ and of the stole of immortality lost by the first parents, are later interpretations rather than clues to the origin.
Its function in practice is clear enough. It is conferred at ordination, in a rite where the manner of its placing enacts the order being conferred. It is worn for every sacramental act, and a priest hearing a confession or anointing the sick in ordinary clothes will still put on a stole, which makes it the one vestment that regularly appears outside a church and away from an altar.
Colour follows the liturgical season as the chasuble does. Many churches now use stoles as the main carrier of design and of local textile traditions, so a stole is often the most individual thing a minister wears.