besides the minimum info on Wikipedia, I am lost on what they do...

I thought we did not discuss explosives, bomb making, drug making etc. on this forum.

I believe the rule says we're allowed to talk about the mechanisms by which such things work, but not to give instructions on how to build one.  Looking at The_Terminator's post, I have no idea how to make a nuclear bomb.

a hydrogen bomb is a fusion bomb.

the nuclear fission of U-235 is used to create high pressure and high temperature to initiate and sustain the chain nuclear fusion reaction of hydrogen.

Simply put, a hydrogen bomb is just a regular fission bomb enhanced by a small bit of fusion.  In reality, the majority of the "oomph" from it comes from the initial fission explosion.  This fission explosion provides enough heat, energy, and pressure to fuse a little bit of the hydrogen contained within the critical mass.  The conditions for fusion to take place only exist for a mere few moments before the dispersal happens which results in the material being spread all over the place.  A good analogy would be a poison tipped bullet.  While the poison at the tip of the bullet adds more lethality, the great bulk of the lethality of the bullet comes from the bullet itself.  

There is actually a second fission stage which results in much of the energy.

First, the (relatively small) fission trigger goes off, releasing large amounts of electromagnetic radiation.
This bounces off the outer casing of the weapon, which is made of U-238.
The radiation is reflected inward, and this generates the conditions (temperature, pressure) to fuse the hydrogen isotopes.
The fusion reaction releases lots of high energy neutrons.  These, unlike those emitted from the fission trigger,
are energetic enough to fission the U-238 of the outer case.  (U-238 normally doesn't fission, it needs something like 1 MeV extra energy besides the new neutron to fission, but the fusion-produced neutrons have this energy).

This second round of fission (of the U-238 casing) is indeed responsible for a big chunk of the total energy release.  (It is also responsible for tremendous amounts of radioactive fallout.)  I don't know any solid percentages, however.

I think I remember a claim that the 'Neutron Bomb' was one of the designs that didn't use the U-238 jacket.
It couldn't, since it was designed not to leave radioactivity, as much as possible.  That supposedly made it harder to design.

Tsar Bomba might be kind of a special case.  It seems like it was really a demonstration, and a not practical weapon.  My guess is that it also used U-238, but didn't depend on it much for yield.

edit: added missing 'not'

so who began this? or is there no set person?

In theory, a fusion bomb has no upper limit. However, how is it possible to sustain a nuclear fusion reaction for a significant period to sustain sufficient fusion reaction?

IIRC these high yield devices are multistage bombs, with mid-stages added to help sustain fusion.

Besides, it is completely different thing to sustain fusion continuously (like in power station) and for a limited period of time (like in bomb).