To start off a discussion of molecular orbital (MO) theory one must appreciate why chemists have come up with such a theory. Chemistry can be considered the study of molecules. How molecules are formed, how they react with each other and most importantly what holds them together? If one can describe mathematically what holds them together then one can understand what makes them break apart and what makes them react with each other. A molecule is just a bunch of atoms stuck together in a very defined fashion. What makes these atoms stick together? - The electrons which are shared between them. Therefore a theory which describes how electrons are distributed in a molecule would be most useful in describing the structure of a molecule. This theory is called molecular orbital theory.
To understand this theory more clearly we must step back and define what an atom is - a positively charged nucleus with electrons around it. Next we need to describe how electrons are distributed in an atom. Motion of electrons around a nucleus can be described by mathematical equations called wave equations - solutions to these equations are called orbitals. When these solutions are plotted out graphically they form various shapes, either spheres or oblong lobes. Each point which compose these orbitals has a positive or negative numerical value (a zero value represents a node). These positive or negative values are represented by labeling areas with a plus or minus sign. These signs do not represent charge! What physical reality do these graphically represented solutions have? The square of each value represents the probability of finding an electron there. Therefore a plot of the probability function will look very similar to the plot of the orbital, only now there are no negative regions. These plots represent the distribution of electrons in an atom (the region of space where an electron most likely is).
This theory of describing the location of an electron around the nucleus (called Wave Mechanics or Quantum Mechanics) can also be utilized to describe electrons in molecules. This description is very important because the electrons in a molecule are the electrons which are shared between the atoms. These electrons are the glue which makes the atoms stay together with defined structure. Electrons which make atoms stick together are referred to as bonding electrons. To describe the region of space where these electrons reside (a bonding orbital), molecular orbital theory was developed which is also based on Wave Mechanics. Linus Pauling developed the description of a bond in the 1930's which is still used today. Since we already have a good way to describe electrons around an atom using atomic orbitals, Professor Pauling thought 'why not just combine these orbitals to form new ones'. This is precisely what is done to form the molecular orbitals in molecular orbital theory. In terms of wave mechanics: bonds are made by the in-phase overlap of atomic orbitals. Orbitals, which are solutions to wave equations, behave much like classical waves when they are combined . They may interact in a reinforcing way - if the overlap is between areas of the orbital which have the same sign. They interact in a destructive way if the overlap is between areas of opposite sign. In-phase overlap leads to a new orbital of lower energy called a bonding molecular orbital. Out of phase overlap between the same two orbitals leads to a destabilizing interaction which gives a new orbital of higher energy called an antibonding molecular orbital. When combining (sometime referred to as mixing) orbitals a general rule to follow is: the number of orbitals in (being mixed) must equal the number of orbitals out (i.e. the number of atomic orbitals being combined must equal the number of molecular orbitals being formed).
As an example consider the overlap or mixing of 2 p orbitals. Dark shaded regions represent regions where the mathematical sign of the orbital is positive.
So, to put it simply, molecular orbitals are formed when atomic orbitals (s, p, sp3 etc.) which overlap are added together. Electrons which were originally in the atomic orbitals now reside in the molecular orbital. Notice that the electron which was in the p orbital was localized on one atom, when it is in a molecular orbital it is delocalized over two atoms - a situation of considerably lower energy.