Matrix is a sci-fi virtual-reality/cyberspace term.

Originally, matrix meant a womb, echoing the womb-like pods Neo and the others are kept in.

The science fiction meaning, however, is more recent, found in Doctor Who's Deadly Assassin (1976) as an equivalent of cyberspace:

How can you intercept thought patterns within the matrix itself?

Or as a VR-world in Gibson's Neuromancer (1984):

He'd operated on an almost permanent adrenaline high ... jacked into a custom cyberspace deck that projected his disembodied consciousness into the consensual hallucination that was the matrix.

And in modern times, a matrix is a common computer science and mathematics term. It is a rectangular array of numbers, and computers operate using matrices in an absurd number of operations (including rendering the pixels in which you are reading these words).

The movie franchise is fairly ambiguous, but intuitively, I've always assumed the matrix represented the computer-world in which humanity now lived. They are trapped in that array of numbers which controls all their inputs and outputs.

One of the definitions in the Merriam-Webster for Matrix is

something within or from which something else originates, develops, or takes form.

from Vocabulary.com

an enclosure within which something originates or develops (from the Latin for womb)

Another definition is:

an array of circuit elements (such as diodes and transistors) for performing a specific function

In the movie, the Matrix generates the fake reality everyone lives in. The humans were physically arranged as an array of batteries in pods resembling a womb to supply power to the machines.

The Wachowski's concept of the matrix is heavily influenced by Baudrillard's description of "hyperreality":

Baudrillard defined "hyperreality" as "the generation by models of a real without origin or reality"; hyperreality is a representation, a sign, without an original referent. According to Baudrillard, the commodities in this theoretical state do not have use-value as defined by Karl Marx but can be understood as signs as defined by Ferdinand de Saussure. He believes hyperreality goes further than confusing or blending the 'real' with the symbol which represents it; it involves creating a symbol or set of signifiers which represent something that does not actually exist, like Santa Claus.

This hyperreality is created by burying the modern observer in images and text that overlap and come between the observer and reality, to the point where they dominate so completely that they become reality, from the observer's perspective. It is helpful to visualize this as a dense mesh or net of images that surrounds the observer and obscures his view.

So the important part(s) of the dictionary definition of matrix may be:

fine material used to bind together the coarser particles of a composite substance

a rectangular array of quantities or expressions in rows and columns that is treated as a single entity and manipulated according to particular rules

...since these elements of the definition add the concept of the matrix as something in which one is embedded or bound, and which is a dense and artificial grid.

Matrix, Matter, Mother. Mother of all. Origin of all creation.

We come from matter. The root word of matter is mater, which means origin, source, mother. It is the root word of matrix, matter, mother, matriarchy, and so on. This is the mother which gives birth to us in our physical world.

The urubobos – the primordial matrix – contains in “embryonic” form everything that can in principle possibly be experienced, and the thing that does the experiencing. The great serpent (the matrix) is therefore consciousness – spirit, before it manifests itself – and matter, before it is separated from spirit.

The matrix of all things is something feminine, like the mothers of experience; is something with an endlessly fecund and renewed (maternal and virginal) nature – is something that defines fertility and, therefore, femininity itself. Things come from somewhere; all things have their birthplace.

...The eternally extant domain of the unknown therefore constitutes the matrix from which all conditional knowledge emerges.

...The ancient Mesopotamian creation myth – the Enuma elish – provides a concrete example of the interplay of these “personalities.” This myth features four main characters, or sets of characters; Tiamat, the feminine dragon of chaos, primordial goddess of creation (the uroboros and the Great Mother are conflated, as is frequently the case, in this myth); Apsu, Tiamat’s husband and consort; the “elder gods,” children of Tiamat and Apsu; and Marduk, sun-deity and mythic hero. Tiamat symbolizes the great unknown, the matrix of the world; Apsu the known, the pattern that makes regulated existence possible. The “elder gods” symbolize the common psychological attributes of humanity (the “fragments or constituent elements of consciousness”), and constitute a more thorough representation of the constituent elements of the “patriarchal” known; Marduk – greatest of the secondary deities – represents the process that eternally mediates between matrix and regulated existence....

The unknown is unexplored territory, nature, the unconscious, dionysian force, the id, the Great Mother goddess, the queen, the matrix, the matriarch, the container, the object to be fertilized, the source of all things, the strange, the unconscious, the sensual, the foreigner, the place of return and rest, the maw of the earth, the belly of the beast, the dragon, the evil stepmother, the deep, the fecund, the pregnant, the valley, the cleft, the cave, hell, death and the grave

• Jordan Peterson, Maps of meaning

Neo was a computer hacker. It’s a common term in Computer Science and Mathematics to describe data in a 2-dimensional array.