What is a halogen burner or (?) cooktop? How does it work?

Halogen hobs are a type of electrical resistive hob which work just like a halogen incandescent bulb. In fact, the reason why incandescent bulbs are being phased out is that they are so inefficient, giving off most of the energy they use as heat instead of light. A halogen hob uses the same kind of tungsten filament as the lamps, instead of the iron found in the widespread electrical hobs.

The benefit is that it warms up quite fast, much faster than iron.

I haven't shopped for resistive cooktops in recent years, so I don't know how the market is. It's possible that they are very rare and expensive, or that all the new ceramic cooktops use them and the manufacturers don't mention it on the label, so the public is unaware of it. They don't work differently than standard burners, just faster, and they don't look much different from the outside. You also don't need any kind of special cookware for them, I have no idea why the manufacturer of that pan decided to tell you that it works on both "electric" and "halogen", when it is the same thing where pan suitability is concerned.

Update as per ChrisH's comment: Because the halogen works mostly with radiant heat, it's important that you don't use a pan with mirroring bottom, that's probably why the pan was labelled as suitable for halogen besides "electric". This is probably important for iron-heating ceramic cooktops too, because the ceramic also lets a significant part of the iron's radiant heat through.