The omophorion is a broad band worn about the shoulders, originally of wool, and the interpretation given in the Byzantine tradition is fixed and unanimous: it is the lost sheep that the shepherd finds and carries home on his shoulders, and the bishop wearing it is carrying the human race that Christ took up. That is not a decorative gloss added later. It governs the material, the wool, and it governs why the vestment belongs to bishops alone.
It is also the vestment that makes the liturgy episcopal. A bishop who is present without it is present as a bishop but does not preside as one, and the older canonical texts are explicit that he does not serve without it. Where a Latin bishop is identified by mitre and crozier, an Orthodox bishop is identified by this.
Two forms are in use. The great omophorion is long and broad, passing round the shoulders with one end falling in front and the other behind, and it is worn for the opening portions of the liturgy. The small omophorion is shorter, with both ends hanging in front, and is worn later in the service and at other rites. In practice a bishop may put on and take off the omophorion a dozen times in a single liturgy, and each removal is a statement rather than a convenience.