The difference between magnetic fields and electromagnetic radiation As a layperson, I understand electromagnetic radiation to be a wave of photons with a certain frequency, and different frequencies have different characteristics to us. For example, we can detect with our eyes certain frequencies, but others we cannot, even if they can cause otherwise visible effects (such as heating something up).
A magnetic field, however, I imagine as something like a "force field" that attracts or repels things, which we can fluctuate to do work such as spin a rotor.
I believe the two are related, but how? What does a wave of photons have to do with attracting things? I am looking at some practical, visual example of their relationship, if possible.
Here is another example that might further demonstrate my confusion: crystal radios are powered by radio waves. Radio waves are often referred to as electromagnetic radiation that is at a low enough frequency that we cannot see it. I can imagine how a fluctuating magnetic field could power a radio and produce sound, but I can't picture how electromagnetic radiation could.
 A: First, a magnetic field does no work. Induced electric fields by a magnetic field( for example a time changing magnetic field), produce work. See the discussion here: physics.stackexchange.com/questions/67826/…  and look for Jim's comments on the accepted answer. Magnetic fields generate electric fields inside an object and that's how work is produce. Although macroscopically it may seem that magnetism is ding all the work it's not. 
And may I ask why you cannot picture electromagnetism doing work? Let's mention first that the electric and magnetic fields do not exist completely alone. By Special Relativity considerations one shows that in an inertial frame were only one of the fields exist, in another both exist.
From the scope of classical electromagnetism( Maxwell equations) it's not so valid to discuss the existence of photons. Electromagnetic radiation is the propagation of time dependent coupled electromagnetic fields- by coupled I mean that one field has a phase relation with the other. This result comes from solving the Maxwell equations, the equations of motion for classic light.This radiation carries energy and momentum in the fields and these fields exert a force on object via a force called Lorentz force (https://en.wikipedia.org/wiki/Lorentz_force and https://en.wikipedia.org/wiki/Covariant_formulation_of_classical_electromagnetism).
So classical light can exert a force, but it's just to small to feel. When the light of the Sun heats you on a hot day, a force is exerted on your body on a molecular level, this force translate in work is the energy your body cells gain and so they warm up. You can also think of visible light of lasers, where a highly energetic beam concentrated can do real damage to an object or the case of gamma radiation that can damage your DNA.
If you now take quantum mechanical considerations you can think of light being quantized, but to do that properly one should quantize the electromagnetic field. A simpler approach is to treat the interacting particles( electrons for example) as quantum mechanical and the interaction between them taken as granted to be the electromagnetic potentials. Anyway, you may consider photons but these particles are not particles in the sense of a concrete classical body, since they describe a field.
Finally, I should mention once more that a body moved from an electric field( of a capacitor) could be seen from another frame of reference as moved by an electromagnetic field. And a body that is moved by magnetic field( a magnet) should be understood as a moving body because of the induced electric fields inside it.
Hope this helps.
A: 
As a layperson, I understand electromagnetic radiation to be a wave of photons with a certain frequency,

According to the classical picture, which is good enough for 90% of the engineering applications, electromagnetic radiation is a wave of oscillating electric and magnetic fields. Photons, on the other hand, are considered as particles and used to explain certain phenomena where classical picture fails. This is all about particle-wave duality and not many people can understand it fully. One can safely accept it as it is.

and different frequencies have different characteristics to us. For example, we can detect with our eyes certain frequencies, but others we cannot, even if they can cause otherwise visible effects (such as heating something up).

Yes, this is right. One can look at the electromagnetic spectrum to see the naming convention for electromagnetic waves with different frequencies.

A magnetic field, however, I imagine as something like a "force field" that attracts or repels things, which we can fluctuate to do work such as spin a rotor.

Well, you can consider an electric field as a force field, too. A positive charge creates a field around it and attracts negative charges and vice versa.
The key point maybe that when we think about electric and magnetic phenomena we think about static events, that is, there is a permanent magnet (or charge) I bring another permanent magnet (or charge) close by and they either repel or attract. Electromagnetic radiation is a dynamic thing. It is all about changing electric and magnetic fields as opposed to static ones.

I believe the two are related, but how? What does a wave of photons have to do with attracting things? I am looking at some practical, visual example of their relationship, if possible.

I think the phrase "wave of photons" is not quite right as I tried to explain above. Photons are particles to represent electromagnetic radiation. They have momentum and energy just like other particles.

Here is another example that might further demonstrate my confusion: crystal radios are powered by radio waves. Radio waves are often referred to as electromagnetic radiation that is at a low enough frequency that we cannot see it. I can imagine how a fluctuating magnetic field could power a radio and produce sound, but I can't picture how electromagnetic radiation could.

I think my above explanations answer this question but I can summarise with some quotations and an image from wikipedia page:
"Classically, electromagnetic radiation consists of electromagnetic waves, which are synchronized oscillations of electric and magnetic fields that propagate at the speed of light through a vacuum... Electromagnetic waves are produced whenever charged particles are accelerated, and these waves can subsequently interact with any charged particles."

A: An electromagnetic field can be viewed as a combination of an electric and a magnetic field.
See for example this youtube video for the classical representation of an EM field as a superposition of an electric and a magnetic field, or the following pictorial representation:

This happens because, due to Maxwell's equations, an oscillating electric field inevitably induces a magnetic field, and vice versa an oscillating magnetic field induces an electric field.
More specifically, this is due to Ampère's and Maxwell-Faraday laws:
$$ \nabla \times \mathbf B = \mu_0 \left( \mathbf J + \epsilon_0 \frac{\partial \mathbf E}{\partial t} \right),
\qquad
\nabla \times \mathbf E = - \frac{\partial \mathbf B}{\partial t},
 $$
which relate the rage of change of $\mathbf E$ and $\mathbf B$ to $\mathbf B$ and $\mathbf E$, respectively.
A travelling electromagnetic wave is possible for this exact reason: an oscillating electric field will induce a change in the magnetic field, which will induce a change in the electric field and so on.
This means that yes, a moving magnet will emit electromagnetic radiation.
You seem to only relate the words "magnetic field" to a magnetostatic field, which is what is produced by a (steady) magnet, and that indeed does not induce an electric field.
