Or more specifically, What physical law can X manipulate in order to affect the electromagnetic force?
For something to base our assumptions off of, lets assume that we can;
An extension of of the previous post: What would powers based of Gravity look like?
As others have observed, almost everything involved in the regular day-to-day interaction with matter involves the electromagnetic force.
The problem is though that many of these interactions are at a very small scale. Materials can't interpenetrate each other in a large part thanks to the Coulomb forces causing the electron clouds around their respective atoms to repel each other. Away from the atomic scale though you simply can't "see" the charges of the electrons, because they are effectively screened by the charges of the protons they're bound to. That's why matter seems neutral at a macroscopic scale. You can apply truly collossal electrical and magnetic fields to overcome this... when you have electrical fields that approach a tenth of a billion volts per meter or magnetic fields of the order of billions of tesla you can do all sorts of weird stuff to matter but the side effect is that you'll destroy pretty much everything in the area of effect and need mindboggling amounts of power to induce such field strengths.
So, you should forget:
What you should be cautious about:
Throwing lightning. Those electrons have to come from somewhere, you know. Conjuring them out of the quantum vacuum is extremely hard and produces a lot of positrons too soo you'll only end up irradiating yourself.
If you what a big enough potential difference between your hands to cause arcing, it involves bits of your own body being ionised and flying through the air. This is going to cause some damage, potentially a lot of damage. If you make yourself immune to ionisation you'll get air breakdown instead, which may or may not be enough to look cool.
Remember that lightning behaves oddly and doesn't always hit the things you expect. Putting a huge potential difference between you and your target isn't necessarily going to involve lightning shootin gout of you and into them... you might both just arc into the scenery, or you might find that the scenery around them is enough to protect them from the effect.
What will probably work just fine:
What might not be practical is the generation of electromagnetic waves, eg. light, radio, etc. This is because they require very, very high frequency changes in an electrical and magnetic field. Being able to generate a magnetic field is not the same as flicking it back and forth at a precise rate a few trillion times a second! The hands are of course yours to wave, but this is quite an important distinction. If I were designing a magic system around this, I'd make radio or light-type EM radiation manipulation entirely distinct from "simple" electrical or magnetic field manipulation.
Almost everything people experience on a daily basis is electromagnetism. Full-blown control over this force is almost indistinguishable from reality-warping. One could plausibly even pull off some weird transmutations of elements, by reducing the charge between protons to get cold fusion.
It's more helpful to narrow things down to specific aspects of the force. If the power is more like generating an arbitrary EM field, and being able to shape or vary it, then you have something like Magneto, including the things he could do that the movies toned down (force-fields, blasts, etc). If you can mess with ions, the number of crazy chemistry tricks available become too numerous and complex to summarize, beyond pointing out that messing with ionic bonds seems like it should enable the weaponizing of table salt.
Can they emit light? X-rays are the most energy-efficient way to accomplish quite a few things, especially if those things involve killing. How about manipulating light, or the ways in which matter interact with photons? You now have both X-ray vision and telescopic / microscopic vision. If you can interpret radio, two people with these abilities are effectively telepathic. Not to mention all the potential for illusions, screwing with communications technology, etc.
When you get down to it, many, many superpowers are effectively a highly restricted form of EM manipulation, or can be accomplished thereby. Without some sort of specific rules, the answer is "almost anything not covered by the other forces, and you can fake some of those, too".
If rules are required to make such a power less god-like, restricting it in terms of scale, magnitude, or specific aspects (Magneto Vs light-bending) can help. These are still fairly overpowered (being able to wipe every hard-drive within a 100m radius, or lethally X-ray everything within a 100m radius are pretty terrifying).
This person, since their powers can affect electromagnetism, would be able to induce variable magnetic and/or electric fields.
Thus it would be possible for them to:
As a side note, every time we interact with the outside world we do it via "electromagnetic power": when I push an apple, the electron cloud in my finger tip interacts with the electron cloud of the apple (Coulomb repulsion, electromagnetic), resulting in the apple moving.
In addition to the answers already posted, what about messing with your brain?
From my limited understanding of biology I believe neurons operate via electrical signals so depending on how precise your control is and how well you understand the neural structure of the brain you could in theory manipulate thoughts by initiating tiny changes in the electrical fields between neurons.
Basically Bedlam from Deadpool 2.
