There are two types of nuclear reactions that can occur to create explosions: fusion and fission.

With fusion, you are forcing two light atoms, such as hydrogen, to make one heavier atom, helium. You have basically discovered the reason why we don't yet have nuclear fusion power plants. It takes a lot of energy to force two charged particles together (like two positively charged hydrogen nuclei, the electrons are actually out the picture). What you have to do is heat up the atoms so much that they are moving fast enough to overcome the electric force. The center of the sun is hot enough that fusion can occur and we receive the excess energy as sunlight.

Nuclear fission occurs when an unstable heavy particle breaks apart spontaneously. You can create a nuclear fission explosion by collecting a critical quantity of fissionable material such as certain isotopes of uranium or plotonium. When some atoms break up they release energy and particles that cause other atoms to break up: a chain reaction that ends with an explosion. Nuclear power plants run by using the energy from fission to heat up water and make steam to run generators. They can control how many atoms break up at any given time to supply the desired amount of energy without getting too hot (also know as meltdown). Nuclear weapons are powered by a fission reaction or fission is used to start a much more powerful fusion reaction.

This is a very good question.In fact, many scientists are currently spending a lot of time thinking about this very question. Your question basically summarizes the challenge for scientists who are trying to get energy out of joining atoms together.

As you already know, all atoms are made up of a positively charged nucleus which contains particles called protons and neutrons as well as negatively charged electrons which circle the nucleus. When two atoms get together, the negatively charged electrons and, even more importantly, the positve charges on each nucleus keep the two atomic nuclei far apart (by far we are comparing the distance between the nuclei to their diameters). For certain light elements, if we apply a lot of energy, we can force the nuclei together, and this releases a lot of energy. The question is how do we apply the energy needed.
There are a few approaches which either have already been used or are currently being worked on:

1. One idea is to put tritium (an isotope of hydrogen) inside the core of a uranium or plutonium based atomic bomb. The explosion generated by the atomic bomb will cause the tritium to fuse together, releasing a lot of energy. This combined device is, of course, known as a hydrogen bomb and is many times more powerful than the uranium device alone. This device is not practical for generating energy in a controlled fashion for say electricity, so scientists are currently working on several other approaches.

2.One idea for getting the controlled release of energy is to put tritium (our isotope of hydrogen) in a glass bead and heat this up to millions of degrees Farenheit using strong lasers. So far, scientists have had to put more energy into this rection with the lasers than they get out. Not very practical! Work is ongoing in this field.

3.Another idea involves heating tritium up in what is known as a plasma reactor.
This is rather complicated, but has some similarities to a neon light. Electric and magnetic fields are used to warm and contain the tritium plasma, and hopefully get it to fuse together. In a plasma, a lot of the tritium has has the electrons separated from the nuclei by the electric and magnetic fields. This removes the problem that you mentioned, but the positively charged nuclei still repel each other, and only very few will combine. So far all plasma reactors that have been built require more energy input than they get out. Again, this is not acceptable if we want to generate electricity.

Now, I have a question for you:

As I said, the positively charged nuclei repel each other; why then will they fuse together and release energy?

Hint: Look in a chemistry or physics book for a discussion of the strong nuclear force.

There are several issues in what you ask -- indeed, the electron clouds surrounding the atoms do repel each other. However, if two atoms are sent towards each other so that they collide, as they approach each other, the electron clouds repel each other, producing a force on the atoms. However, Newton's laws predict that there must also be an equal force between the electrons and the atom since the majority of the atom's mass (and hence momentum) are carried in the nucleus. So, the repulsion causes the clouds to deform and they will no longer be symmetric around the nucleus, but will prefer to point away from the other atoms electrons.

At high enough collision energy, the electrons give up -- i.e. they are ionized from the atom rather than prevent collision. (This ionization occurs at relatively low energies for the first few electrons. -- Consider the "electron gun" present in you T.V. pisture tube). However, the nuclei are themselves positively charged. -- They too repel each other. So even higher collision velocities are needed.

Finally, if the two nuclei are close enough, the strong force (the one that hold atoms together) may win the battle, and the atom will fuse. For this to occur, the nuclei must be very close indeed since the stong force is effective only over very small distances. At this point the electrons from both atoms are left hovering around the new fused atom, which often undergoes several forms of energy loss to make a (relatively) stable nucleus. For very light atoms (H, D, He, Li ..) this can be done on a massive scale by using a fission atom bomb to provide the initial collision velocity. The technique has also been done in large accellerators to make new super massive atoms (e.g. atoms heavier that Uranium -- e.g. Cf, Pt, etc.)

So technique is in a sense as simple as possible -- put a lot of pressure and heat on the material so that its atoms collide at high velocities. In practice, the energy needed to do the smashing is very high indeed. Even on the Sun --the region of actual fusion of Hydrogen to Helium is located only at the core in almost unimaginable pressures and temperatures. The pressure at the core of Jupiter fell short of this -- and so Jupiter is a planet and not a star...

Ionization of electrons from atoms is one of the key physical operations in many chemical reactions -- can you think of systems which can preform the trick at low energies and with very little pressure? Hint --think green...