Noah's messenger of a world made new, the form 'like a dove' at the Jordan, the poor man's temple offering — and, in later shorthand, peace itself.
Andrea del Verrocchio, Leonardo da Vinci - Baptism of Christ - Uffizi.jpg — Andrea del Verrocchio / Leonardo da Vinci
Origin
Biblical: the dove returns to Noah with an olive leaf as the flood recedes (Gen 8:11); doves are the offering of the poor — Mary's offering (Lev 12:8; Luke 2:24); the Song of Songs makes the dove tenderness itself; and at Jesus' baptism the Spirit descends 'like a dove' (all four Gospels). Pagan antiquity also used doves (notably for Aphrodite/Venus — stated plainly), which the church neither borrowed nor feared; the biblical dove had its own pedigree.
Biblical references: Genesis 8:8–12 · Leviticus 12:8 / Luke 2:24 · Matthew 3:16 (par. Mk 1:10; Lk 3:22; Jn 1:32) · Matthew 10:16 · Song of Songs 2:14
Meaning by Tradition
Jewish
Israel itself is the dove (Hosea 7:11; rabbinic readings of Song of Songs — tradition, not Scripture); Noah's dove is the herald of a judged and renewed world.
Early Church
Baptismal art's fixture: the dove descends over Christ in the Jordan in the earliest baptistery programs (Ravenna's two baptisteries). On epitaphs, the dove with olive branch and 'IN PACE' marks a soul at peace.
Orthodox
The Spirit's self-disclosure at Theophany — the feast that reveals the Trinity: the Father's voice, the Son in the water, the Spirit as a dove. Eucharistic doves (vessels for the reserved gifts) hung over some ancient altars.
Catholic
The standard image of the Holy Spirit in art and architecture (Bernini's window in St. Peter's); also of the soul's peace and, popularly, of peace among peoples.
Protestant
Kept close to the texts: the Spirit's descent at the Jordan and Jesus' 'wise as serpents, innocent as doves'. The secular peace-dove is acknowledged as a borrowing from Noah's bird.
Three doves, one thread
Noah's dove announces a world coming up out of the waters of judgment. At the Jordan, as Jesus comes up out of the water, the Spirit descends like a dove — the evangelists could hardly signal 'new creation' more clearly. And the dove was the offering of the poor: the bird of new beginnings was also the bird anyone could afford.
Like a dove
The Gospels say the Spirit descended 'like' (hōs/hōsei) a dove — a visible descent in dove-like form or manner. The church's art made the simile a figure; theology should remember the Spirit is not a bird. The image conveys gentleness, anointing, and the hovering of Genesis 1:2 (the same Spirit over new waters).
Pastoral Caution
The dove has been so absorbed into generic 'peace' branding that its scriptural referents — judgment survived, the Spirit's anointing, the poor's offering — go unheard. Restore the texts and the bird becomes weighty again.