If like charges repel each other, why do electrons of the same (or different) element attract while forming a covalent bond?

 

 

 The electrons have a negative charge which produces an electrostatic force of repulsion.
But, the electrons keep spinning, which produces a magnetic field. This means the electron is like a magnet, with a north and a south pole. The most stable situation is for the direction of the spin of one electron to be the opposite of the direction of the spin of the other electron, which produces a magnetic force of attraction. 

So, the electron pairs exist as the magnetic force of attraction is stronger than the electrostatic force of repulsion, whenever a pair of electrons exists.