There is no simple explanation; I could discuss quantum mechanics, which is the physical basis for chemistry, but that's quite complicated indeed.

How does quantum mechanics enter into this? Electrons don't actually orbit the nuclei of atoms. Instead,electrons are located in probability distributions around atoms called electron shells. All atoms have the same electron shells, but since different elements have different electric charges, the shells filled, is different in each element. When two elements bond together, a new "orbital" becomes more thermodynamically feasible (i.e. it releases energy to go there), and the electrons move into this new orbital that involves both atoms. It now becomes difficult to pull the atoms apart, because doing so would require pulling the electrons out of the potential well.

Different bonds between different elements are stronger or weaker, of course. Elements needing only a few electrons to fill their shells (e.g. oxygen) will just take electrons from atoms that have a few leftover from their outer shells (e.g. metals). That's why certain chemicals, when put together, react.

When two things combine it is normally because the combined material is preferred from an energetic point of view, which means that the new product is more stable than the individual components. The rearrangement of the components into a new product is because electrons and atoms can move all the time and look for a more favorable state.

Atoms have three main parts: protons, neutrons and electrons. The protons and neutrons hang out inside the nucleus of the atom, while the electrons orbit around the nucleus. Electrons are divided into separate shells outside the nucleus, and there are only a certain number of electrons allowed in each shell. Each atom has one or more different configurations of these parts. Some configurations allow for more electrons in a shell, and some can spare an electron, and that is how two atoms can come together.

This can also occur with molecules, where one atom in a molecule has an electron that it will share with an atom from another molecule. This is called Orbital Theory and will be discussed in your chemistry class.