A profession, or as we usually say, a vocation (1 Corinthians 7:20). The "hope of your calling" in Ephesians 4:4 is the hope resulting from your being called into the kingdom of God.
Source: Easton's Bible Dictionary
Calling
CALLING, n.
1. A naming, or inviting; a reading over or reciting in order, or a call of names with a view to obtain an answer, as in legislative bodies.
2. Vocation; profession; trade; usual occupation, or employment.
Pope. Swift. 1 Cor. 7:20
3. Class of persons engaged in any profession or employment.
4. Divine summons, vocation, or invitation.
Give all diligence to make your calling and election sure. 2 Pet. 1.