I am going to start my answer by referring to a different one: What is more fundamental, fields or particles?

The photon is really just a special case of what is outlined in this answer. Quoting DanielSank:

Consider a violin string which has a set of vibrational modes. If you want to specify the state of the string, you enumerate the modes and specify the amplitude of each one, eg with a Fourier series

$$\text{string displacement}(x) = \sum_{\text{mode }n=0}^{\infty}c_n \,\,\text{[shape of mode }n](x).$$

The vibrational modes are like the quantum eigenstates, and the amplitudes $c_n$ are like the number of particles in each state. With that analogy, the first quantization notation, where you index over the particles and specify each one's state, is like indexing over units of amplitude and specifying each one's mode. That's obviously backwards. In particular, you now see why particles are indistinguishable. If a particle is just a unit of excitation of a quantum state, then just like units of amplitude of a vibrating string, it doesn't make any sense to say that the particle has identity. All units of excitation are the same because they're just mathematical constructs to keep track of how excited a particular mode is.

A better way to specify a quantum state is to list each possible state and say how excited it is.

A photon is exactly that: a unit of excitation ⁽¹⁾ of a mode of the electromagnetic field.

The main problem with the photon is that people try to over trivialise it. This has roots in history. In the early days of quantum mechanics particles and in particular photons were invoked to explain the "particle features of light". In the modern view of quantum field theory this picture gets replaced by what DanielSank describes in the linked question.

As such a photon is complicated. It is not a priori a wavepacket or a small pointlike particle. The field theory unifies both these pictures. Real photon wavefields then are superpositions of these fundamental excitations and they can display both field and particle behaviour. The answer to the OPs following question...

How can we understand pair production if we don't understand what the photon is? Or the electron? Or the electromagnetic field? Or everything else?

...lies therein. If you want to know what happens to the real physical objects, you are moving away from photons. Single photon states in nature are rare if not non-existent.

So what is a photon fundamentally?

Quoting from the question:

[...] "the photon is an excitation of the photon field". That tells me nothing.

It tells a lot, the mathematical formalism is very clear and many people have tried to explain it in answers here and elsewhere.

[...] because it gives the impression that photons are forever popping into existence and flying back and forth exerting force. This concept is there in the photon Wikipedia article too. It isn't true. As as anna said virtual particles only exist in the mathematics of the model. So, who can tell me what a real photon is? [...]

The problem here is really the relation between the mathematical formalism and "reality". A "real" photon is not a thing, the photon is a mathematical construct (that was described above) and we use it (successfully) to describe experimental outcomes.

^{(1) courtesy of DanielSank.}

I am going to start my answer by referring to a different one: What is more fundamental, fields or particles?

The photon is really just a special case of what is outlined in this answer. Quoting DanielSank:

Consider a violin string which has a set of vibrational modes. If you want to specify the state of the string, you enumerate the modes and specify the amplitude of each one, eg with a Fourier series

$$\text{string displacement}(x) = \sum_{\text{mode }n=0}^{\infty}c_n \,\,\text{[shape of mode }n](x).$$

The vibrational modes are like the quantum eigenstates, and the amplitudes $c_n$ are like the number of particles in each state. With that analogy, the first quantization notation, where you index over the particles and specify each one's state, is like indexing over units of amplitude and specifying each one's mode. That's obviously backwards. In particular, you now see why particles are indistinguishable. If a particle is just a unit of excitation of a quantum state, then just like units of amplitude of a vibrating string, it doesn't make any sense to say that the particle has identity. All units of excitation are the same because they're just mathematical constructs to keep track of how excited a particular mode is.

A better way to specify a quantum state is to list each possible state and say how excited it is.

A photon is exactly that: a unit of excitation ⁽¹⁾ of a mode of the electromagnetic field.

The main problem with the photon is that people try to over trivialise it. This has roots in history. In the early days of quantum mechanics particles and in particular photons were invoked to explain the "particle features of light". In the modern view of quantum field theory this picture gets replaced by what DanielSank describes in the linked question.

As such a photon is complicated. It is not a priori a wavepacket or a small pointlike particle. The field theory unifies both these pictures. Real photon wavefields then are superpositions of these fundamental excitations and they can display both field and particle behaviour. The answer to the OPs following question...

How can we understand pair production if we don't understand what the photon is? Or the electron? Or the electromagnetic field? Or everything else?

...lies therein. If you want to know what happens to the real physical objects, you are moving away from photons. Single photon states in nature are rare if not non-existent.

So what is a photon fundamentally?

Quoting from the question:

[...] "the photon is an excitation of the photon field". That tells me nothing.

It tells a lot, the mathematical formalism is very clear and many people have tried to explain it in answers here and elsewhere.

[...] because it gives the impression that photons are forever popping into existence and flying back and forth exerting force. This concept is there in the photon Wikipedia article too. It isn't true. As as anna said virtual particles only exist in the mathematics of the model. So, who can tell me what a real photon is? [...]

The problem here is really the relation between the mathematical formalism and "reality". A "real" photon is not a thing, the photon is a mathematical construct (that was described above) and we use it (successfully) to describe experimental outcomes.

^{(1) courtesy of DanielSank.}

I am going to start my answer by referring to a different one: What is more fundamental, fields or particles?

The photon is really just a special case of what is outlined in this answer. Quoting DanielSank:

Consider a violin string which has a set of vibrational modes. If you want to specify the state of the string, you enumerate the modes and specify the amplitude of each one, eg with a Fourier series

$$\text{string displacement}(x) = \sum_{\text{mode }n=0}^{\infty}c_n \,\,\text{[shape of mode }n](x).$$

The vibrational modes are like the quantum eigenstates, and the amplitudes $c_n$ are like the number of particles in each state. With that analogy, the first quantization notation, where you index over the particles and specify each one's state, is like indexing over units of amplitude and specifying each one's mode. That's obviously backwards. In particular, you now see why particles are indistinguishable. If a particle is just a unit of excitation of a quantum state, then just like units of amplitude of a vibrating string, it doesn't make any sense to say that the particle has identity. All units of excitation are the same because they're just mathematical constructs to keep track of how excited a particular mode is.

A better way to specify a quantum state is to list each possible state and say how excited it is.

A photon is exactly that: a mode of the electromagnetic field.

The main problem with the photon is that people try to over trivialise it. This has roots in history. In the early days of quantum mechanics particles and in particular photons were invoked to explain the "particle features of light". In the modern view of quantum field theory this picture gets replaced by what DanielSank describes in the linked question.

As such a photon is complicated. It is not a priori a wavepacket or a small pointlike particle. The field theory unifies both these pictures. Real photon wavefields then are superpositions of these fundamental excitations and they can display both field and particle behaviour. The answer to the OPs following question...

How can we understand pair production if we don't understand what the photon is? Or the electron? Or the electromagnetic field? Or everything else?

...lies therein. If you want to know what happens to the real physical objects, you are moving away from photons. Single photon states in nature are rare if not non-existent.

So what is a photon fundamentally?

Quoting from the question:

[...] "the photon is an excitation of the photon field". That tells me nothing.

It tells a lot, the mathematical formalism is very clear and many people have tried to explain it in answers here and elsewhere.

[...] because it gives the impression that photons are forever popping into existence and flying back and forth exerting force. This concept is there in the photon Wikipedia article too. It isn't true. As as anna said virtual particles only exist in the mathematics of the model. So, who can tell me what a real photon is? [...]

The problem here is really the relation between the mathematical formalism and "reality". A "real" photon is not a thing, the photon is a mathematical construct (that was described above) and we use it (successfully) to describe experimental outcomes.

I am going to start my answer by referring to a different one: What is more fundamental, fields or particles?

The photon is really just a special case of what is outlined in this answer. Quoting DanielSank:

Consider a violin string which has a set of vibrational modes. If you want to specify the state of the string, you enumerate the modes and specify the amplitude of each one, eg with a Fourier series

$$\text{string displacement}(x) = \sum_{\text{mode }n=0}^{\infty}c_n \,\,\text{[shape of mode }n](x).$$

The vibrational modes are like the quantum eigenstates, and the amplitudes $c_n$ are like the number of particles in each state. With that analogy, the first quantization notation, where you index over the particles and specify each one's state, is like indexing over units of amplitude and specifying each one's mode. That's obviously backwards. In particular, you now see why particles are indistinguishable. If a particle is just a unit of excitation of a quantum state, then just like units of amplitude of a vibrating string, it doesn't make any sense to say that the particle has identity. All units of excitation are the same because they're just mathematical constructs to keep track of how excited a particular mode is.

A better way to specify a quantum state is to list each possible state and say how excited it is.

A photon is exactly that: a unit of excitation ⁽¹⁾ of a mode of the electromagnetic field.

The main problem with the photon is that people try to over trivialise it. This has roots in history. In the early days of quantum mechanics particles and in particular photons were invoked to explain the "particle features of light". In the modern view of quantum field theory this picture gets replaced by what DanielSank describes in the linked question.

As such a photon is complicated. It is not a priori a wavepacket or a small pointlike particle. The field theory unifies both these pictures. Real photon wavefields then are superpositions of these fundamental excitations and they can display both field and particle behaviour. The answer to the OPs following question...

How can we understand pair production if we don't understand what the photon is? Or the electron? Or the electromagnetic field? Or everything else?

...lies therein. If you want to know what happens to the real physical objects, you are moving away from photons. Single photon states in nature are rare if not non-existent.

So what is a photon fundamentally?

Quoting from the question:

[...] "the photon is an excitation of the photon field". That tells me nothing.

It tells a lot, the mathematical formalism is very clear and many people have tried to explain it in answers here and elsewhere.

[...] because it gives the impression that photons are forever popping into existence and flying back and forth exerting force. This concept is there in the photon Wikipedia article too. It isn't true. As as anna said virtual particles only exist in the mathematics of the model. So, who can tell me what a real photon is? [...]

The problem here is really the relation between the mathematical formalism and "reality". A "real" photon is not a thing, the photon is a mathematical construct (that was described above) and we use it (successfully) to describe experimental outcomes.

^{(1) courtesy of DanielSank.}