How do Electrons in a Crystal Lattice form Pair bonds? How do electrons around adjacent atoms in a crystal lattice form pair bonds when they are both negative and should repel each other?
 A: Strictly speaking, the bonds are not formed between the electrons, but between atoms, which are electrically neutral, because they also contain positively charged nuclei. It is difficult to give a deeper answer, since there exist many types of chemical bonds and the answer is different for each of them. A good start could be reading about covalent bond in the hydrogen molecule, typically discussed in QM textbooks.
Update
The bond types differ from one type of crystal to another, depending on the type of atoms and the crystal structure. For example, in diamond, silicon, GaAs, all the bonds are covalent, like in organic chains. Although in GaAs they are polar covalent, since the nearest neighbors are different types of atoms (otherwise GaAs and diamond have the same crystal structure). In graphite, the bonds are covalent within each graphene layer, but the coupling between the layers is via van-der-Waals forces. Metals are usually held together by metallic bonds, where electrons are shared by all the ions - one can think of it as a generalization of a covalent bond to more than two atoms (i.e. N electrons are shared by N ions).
