In the LED, is the medium simply the solid component of the LED itself??
If I understand what I think you're asking, yes. Note that most LED's have a lens above them, the lit part itself can be very small. (Doesn't have to be, though.)
Also, there is no spark (electrons jumping across a gap) with this, correct?
Right. No spark, very very little heat. Safer, from a fire perspective.
Finally, do you need DC current for an LED?
Technically, no - But an LED, like any diode, will only illuminate when voltage is applied in one direction. If you were to plug an LED into your wall - Well, OK, maybe through a transformer, call it 60Hz AC at 6 Volts peak to peak - It would blink 60 times per second, and be illuminated just slightly under half of the time.
Visualize the sine wave of the electricity, going from +3V to -3V, starting at the origin (0 V, and in an upward direction). There is a "diode drop" across a diode, which IIRC is around 0.7V for a normal diode, and 1.0V for an LED. Until the power is at +1 V, the diode will not be lit. From the time the wave reaches +1V, through the peak at 3V, and back down to +1V, it will be lit. From the time it drops below +1, through 0, to -3, back up through 0 to +1, the LED will not be lit. At 60Hz, you'll see it flickering. In fact, we had some cheap LED Xmas lights this year that did that. Ugh.
So, for lighting purposes, DC is better. Incandescents don't have much flicker because they're heated up pretty much the whole time, with the peaks of the sine wave "recharging" the heat for when the 0 crossings happen. Fluorescents generally don't flicker, especially on a newer bulb and a fixture with a modern ballast, but they can. LED's on DC won't flicker at all.