I remember hearing about "Field Momentum" - the idea that the electromagnetic fields (e.g. radiation) could have momentum and not understanding how that could be possible. Particles / matter have momentum. I've seen the equation and heard that photons have momentum, but did not understand. But reading today I remembered / rediscovered some understanding.
If you have 2 particles (A and B), uncharged, then you can completely describe the state of the system by the position and momentum of each particle.
However, if the particles are charged, you need more information. Imagine that a and b are not close to each other and are moving slowly. Since these particles are charged, their motions affect each other. The mechanism is through the electromagnetic field - as particle A moves, the electromagnetic field changes. This change propagates away from particle A at the speed of light. Particle B is affected by the motion of A sometime later depending upon its distance away from A. So if you have a snapshot of the position and momentum of A and B, there could still be "information in transit" from A to B (or vice-versa) that has not yet affected their momentum/position, and you need to capture that information in order to fully describe the system.
Field momentum captures this information, and it tries to do it in a way that is similar (symmetric) with the way it is captured for the particles.
Example 1 - in the snapshot, A and B are slow moving and far apart. Prior to the snapshot they were also slow moving and far apart. A and B continue their slow motions.
Example 2 - A and B are slow moving in your snapshot, but immediately prior to the snapshot A was oscillating rapidly. Not captured by the position and momentum of A and B are the electromagnetic waves that currently propagating from A to B. Without these waves, A and B would continue as in example 1 (slow regular motion) - which would be incorrect, because when the waves reach B, B will begin to oscillate (perhaps rapidly if the mass to charge ratio of B is similar to A).