Fire is hot because it burns things like wood and gas that have a lot of energy. The energy of wood and gas is chemical energy, from molecules that have a lot of carbon and hydrogen atoms in them. When these molecules react with oxygen, they get changed into new molecules made up of the same atoms that have carbon and hydrogen and oxygen atoms in them. The new molecules don't have very much chemical energy. But the energy is still somewhere, because it can't be destroyed. The extra energy is in heat.

A nuclear bomb is a special unusual case where atoms actually get destroyed, which gives off HUGE amounts of energy plus new atoms that don't have as much energy as the old atoms. You can search for things like this on www.google.com. (good Internet search engine)

That's a good question, because they are similar in a way.Whenever you change matter (any kind of stuff) it either requires some added energy, or gives off extra energy. When water changes to ice, it gives off a little energy. It's still the same stuff and the same combination of elements. It has just changed from a liquid to a solid.
When you change wood to ashes, it gives off so much energy that you can feel it as heat. In this case, the ashes are completely different from the original wood with different combinations from elements, and can not be changed back like ice to water.

A nuclear bomb involves another kind of change. It involves changing a single element itself by breaking atoms of that element apart. This kind of change gives off a really huge amount of energy in the form of an explosion.

Fire is the result of a chemical reaction between a fuel and oxygen. When the fuel is carbon based (like gasoline, etc.) We can generically write this down by the following:

Fuel + oxygen -> carbon dioxide + water

One way to think about this is by the principle of conservation of energy. To first think about this, we must first realize that all molecules have an associated energy with them, where the amount of that energy is different for different molecules. What this means is that in any given process the total energy is still the same. However, the principle does not say anything about the form the energy has to be in. If we look at the above reaction, we would find that the energy of the products (the CO2 and water) have a lower energy of fuel and oxygen. However, since energy must be conserved, the extra energy is given off as heat. It's that release of energy from the chemical reaction that makes fire hot.

There are two basic types of nuclear bombs - fission and fusion. Atomic bombs are based on nuclear fission. An atomic bomb is primarily made up of the isotope plutonium 239. If a neutron hits plutonium nuclei, it may start the fission process where the nuclei is changed, and some energy and another neutron are produced. The neutron acts as a catalyst since it starts the reaction but as one is taken, another is produced. When this reaction goes on and on, it is called a "chain reaction." Plutonium in itself does not become dangerous until it reaches the point of "critical mass" - meaning it is relatively stable until the density of nuclei is high enough such that neutrons can start a chain reaction. However, if there is not enough mass, the process will quickly die out. Surrounding the plutonium are many explosive charges. To detonate an atomic bomb, the explosive charges are simultaneously fired and the explosion compresses the plutonium into a smaller area. Now, when a neutron is added, the nuclei are close enough such that the fission process can proceed in a chain reaction. As the nuclei are split, the result is an incredible release of that atomic energy. The bomb dropped on Nagasaki was an atomic bomb.

The other type of bomb is a hydrogen bomb, or a fusion bomb. Here, hydrogen nuclei are forced together such that they actually fuse and form a helium nucleus. In this process, there is again a huge release of energy from the reaction. As it is difficult to get the hydrogen atoms to fuse, an atomic fission bomb can be used to get the hydrogen nuclei close enough together to initiate the process. To give you an idea of how much energy this type of reaction release, we can think of the sun. This type of hydrogen fusion reaction is the same process by which the sun (and other stars) generate their heat and light.

Fire is hot because chemical bonds are being broken between atoms in a material and in that process some energy is released as heat (infrared radiation) as well as visible light.

In a nuclear reaction, individual atoms such as hydrogen can, under the right circumstances, combine to form heavier atoms, such as helium, and release a lot of energy. This is called fusion.

It turns out that you can't fuse atoms with more protons than iron and get energy out. On the other hand, you can break apart a heavy atom, such as uranium or plutonium and get some energy.
This is called fission.

Conventional bombs and dynamite make explosions by creating a fire that burns all the fuel very quickly. A nuclear bomb is the equivalent using nuclear energy instead of chemical energy. The reaction is started and all the nuclear fuel "burns" at once, causing a big explosion. A fission bomb is created by getting a critical mass of fissionable material in a small volume. As an atom decays, it releases energy and particles which cause more atoms to decay in a chain reaction. A more powerful fusion bomb is created by using a fission reaction to heat up some radioactive hydrogen to hot enough temperature to be able to fuse together and thereby release a lot of energy.

As an aside, I believe nuclear power plants work by using a controlled nuclear fission "fire" to turn water into steam to drive an electric generator. We don't have a fusion nuclear power plant yet because it is very difficult to start a controlled and sustained fusion reaction. Scientists are working on it though. It would be very nice to have fusion plants as there should be little or no nuclear waste to deal with. We know it's possible; nuclear fusion is what powers the Sun.