In terms of things which are tangible, electromagnetic charge is a property which causes a particle to be either attracted or repelled by another particle. Electricity and magnetism are interchangable and the phenomenon is basically the same as magnets either attracting one another or repelling one another depending on which of their ends are being aligned. Basically, if two particles repel one another, they are said to have the same charge. If they attract, then the charges are said to be opposite.
In atoms, electrons are attracted to protons because they have opposite charges. Unlike magnets however, the two particles don't actually fuse together. There are nucleur forces which keep subatomic particles separated unless they are very close together. The electrons therefore orbit the protons balanced between attractive electromagnetic forces and nucleur forces which keep the particles separated.
Protons and electrons are very different in terms of mass, but they have opposite and equal charges. Therefore, if an atom has an equal number of protons and electrons, the atomic system as a whole has no charge. However, atoms can gain and loose electrons and this imbalance can lead to the atom having either a positive or negative charge.
In atoms with more than one electron, the electrons arrange themselves around the nucleus in a number of orbits. They don't all have the same orbit because they repel one another, since electrons have the same charge. However, two electrons can occupy the same orbit because of something known as spin. If the spin of a pair of electrons is opposite, they both can exist in the same orbit.
The orbits are arranged into shells. Because of a phenomenon known as entropy, everything in the universe naturally seeks to attain its lowest energy level. All the electrons in a shell are repelling one another to certain degrees. This raises the energy of the shell. However, if the shell is filled with the maximum number of electrons it can hold, then a symmetry is generated which lowers the energy of the system. This is the underlying reason atoms can be positive or negative.
Atoms can have many electronic orbitals arranged in a series of shells. The inner shells are filled and stable, but the last shell usually is not filled. Some atoms like Sodium have a single electron in their last shell. The shell can hold many more electrons before it is filled, but it is easier for Sodium to just loose the single electron, and in doing so, aquire a positive charge. This is because it now has 1 more proton than the remaining electrons. Chlorine has its last shell 1 electron short of being filled. The atom will aquire an additional electron to fill its last shell, and become a negative ion, because now Chlorine has 1 more electron than all the protons it has.
If an atom allready has a partially filled shell, it will share its electrons with other atoms to fill its last shell, forming chemical bonds. Because it is sharing electrons, atoms like Carbon do not form positive or negative ions like Sodium and Chlorine. Carbon fills its shell by forming 4 electron pairs. One electron in the pair is its own and the other electron comes from another atom. This is essentially what a chemical bond is. Finally, there are atoms whose last shell is completely filled. these atoms do not need to loose, aquire or share electrons. They are neutral and do not form chemical bonds. Neon is such an atom and is chemically inert.