All nuclear bombs involve fission. Some (termed "thermonuclear") involve fusion. A fission first stage is detonated to ignite a (much higher yielding) fusion second stage. Most bombs deployed today are thermonuclear simply because it's the most sensible way to scale up the yield of a weapon.

Nitpick, AFAIK (which is t far) the real crust-busters (e.g. Tsar Bomba) use yet a third stage of fissile material that gets rapidly burnt up at truly absurd neutron densities to get that last 40-90 Mt and as a result are dirty as all hell. The Soviets turned that weapon down from 100Mt designed yield to 50 because there was no way to keep it from just radiating everything.

Isn’t the fissile material the only reason there is residual radiation/fallout from fusion devices?

A lot of materials are fissile in the presence of a fusion reaction.

This makes sense, thank you

Usually if it's called an "atom bomb" or "A-bomb" it's pure fission, and a nuclear bomb, thermonuclear bomb, or hydrogen bomb is fission-fusion-fission. A fission bomb compresses the hydrogen to cause it to fuse, and the extra energy and neutrons from the hydrogen fusion cause a whole lot of additional fission in the uranium tamper. For details look up the "Teller-Ulam" design, wikipedia has some good descriptions.

I'm curious what you think the hydrogen in a hydrogen bomb would split into.

Sheesh, tough crowd.

We actually got to fusion bombs pretty quickly, in the 1950s. Presumably almost all practical nuclear weapons since then are fission bombs

They're two stage: fission primary and fusion secondary.

Three stage: the fusion secondary typically uses a uranium pusher, which undergoes fission from the fusion neutrons.

In theory you could do a fusion bomb as a stage, that would make be insane. At least if I understand that correctly.

I thought the fusion produces a lot of neutrons that then cause a ton of more fission.

Nope, the most powerful bombs use both - in effect they use fission to perform fusion. Obviously the problem with this for energy production is that it is not a very controlled reaction.

And then they use the neutrons from fusion to cause more fission.

they have a fission primary stage to ignite the fusion secondary stage