In The Matrix Revolutions, when Morpheus, Trinity and Seraph enter the Merovingian's club, one of the guards calls Seraph "Wingless". Where does this name come from?
As his name suggests, Seraph is believed to be a Seraphim (a type of program that performed the function of Agents in the Paradise Matrix). Seraph used to work for the Merovingian but was exiled along with the Merovingian and his other henchmen.1 Eventually he came to protect the Oracle, thus betraying the Merovingian2 and losing his exile status (since he had a purpose again).
At some point Seraph lost his wings, possibly from an attack by the Merovingian similar to the latter's attack on the Oracle which forced her to change her shell. He can no longer fly, in contrast to the guards inside Club Hel who work for the Merovingian and can defy gravity -- they are fallen angels (in Hell no less!) who still have their "wings" (although we can't see them).
(You can learn more about the history of Seraph, the Oracle, and the Merovingian here.)
1We know this because Smith tells Seraph in The Matrix Revolutions that "chasing you was like chasing a ghost", and it is part of Agents' responsibility to apprehend exile programs.
Name aside, Matrix Revolution shows us that Seraph is a Program:
If you look at the colour representation in Revolutions, gold = hardware, and green = software. Just from these observations it can be said that Seraph is not an operating system (Matrix) program which is green, but a hardware program, firmware, which is gold. That is why he is virtually unbeatable, because programs running in hardware are always faster and more efficient than programs running on the operating system.
Furthermore, Matrix Revolution introduces us to programs from the previous Matrix, left overs with non-human-like powers. The Ghost Twins, Cain & Able (Vampire and Werewolf). These were saved from the previous Matrix versions, smuggled in.
Seraph is a protector, dressed in white, and can take on Agents at the same level. This invokes an Angel representation vs the Demons that are Agents
And at the end, In Matrix Online is much more blunt about Seraph:
In The Matrix Online, Seraph was speculated to have been originally a Seraph, exiles that were former Agents of one of the previous Matrices. He became employed by the Merovingian when he was exiled into the Matrix, and had his wings torn out due to betrayal.
In the Game, there are Programs that have an Angelic appearance including wings:
Seraphim are the predecessors of the modern Agents in the current Matrix, They were used in the Matrix betas. They have the general appearance of agents with square glasses, but have wings. Seraph is rumored to have been formerly a Seraphim. Belial was believed to be a Seraphim. Baelroth was a fallen angel exile, presuming that he was once a former seraphim.
There is much of the previous Matrix lore introduced in Revolutions that aren't touched on. Seraph previously being an Angelic Agent that escaped into the new Matrix is possible, and why he is called Wingless is self evident in that respect. He is entering the Merovingian's club, aptly called Club Hel. Where else would an Angel go from falling from grace.
Since the Matrix introduces a lot of imagery that falls under standard Juedo-Christian themes, the evil red burning Agent Smith, the angelic view of Neo in machine code, the savior complex, etc, Seraph is plainly meant to be a Fallen Angel.
His name means angel.
an angelic being, regarded in traditional Christian angelology as belonging to the highest order of the ninefold celestial hierarchy, associated with light, ardor, and purity.
Above him stood the Seraphim; each had six wings; with two he covered his face, and with two he covered his feet, and with two he flew." (Isaiah 6:1–3)
It is possible the Merovingian or someone nicknamed him "Wingless" because he obviously has none, let alone six.
This is how the Bible describes a Seraph:
I saw the Lord sitting upon a throne, high and lifted up; and His train filled the Hekhal (sanctuary). Above him stood the Seraphim; each had six wings; with two he covered his face, and with two he covered his feet, and with two he flew. (Isaiah 6:1–3)
Obviously, the Seraph in the Matrix differs from the usual depiction in that he does not have wings.