Spin states (d electrons)

Spin states when describing transition metal coordination complexes refers to the potential spin configurations of the central metal's d electrons. In many these spin states vary between high-spin and low-spin configurations. These configurations can be understood through the two major models used to describe coordination complexes; crystal field theory and ligand field theory, which is a more advanced version based on molecular orbital theory.
High-spin vs. low-spin.
Octahedral complexes.
The Δ splitting of the "d" orbitals plays an important role in the electron spin state of a coordination complex. There are three factors that affect the Δ: the period (row in periodic table) of the metal ion, the charge of the metal ion, and the field strength of the complex's ligands as described by the spectrochemical series.
In order for low spin splitting to occur, the energy cost of placing an electron into an already singly occupied orbital must be less than the cost of placing the additional electron into an e"g" orbital at an energy cost of Δ. If the energy required to pair two electrons is greater than the energy cost of placing an electron in an e"g", Δ, high spin splitting occurs.
If the separation between the orbitals is large, then the lower energy orbitals are completely filled before population of the higher orbitals according to the Aufbau principle. Complexes such as this are called "low-spin" since filling an orbital matches electrons and reduces the total electron spin. If the separation between the orbitals is small enough then it is easier to put electrons into the higher energy orbitals than it is to put two into the same low-energy orbital, because of the repulsion resulting from matching two electrons in the same orbital. So, one electron is put into each of the five "d" orbitals before any pairing occurs in accord with Hund's rule resulting in what is known as a "high-spin" complex. Complexes such as this are called "high-spin" since populating the upper orbital avoids matches between electrons with opposite spin.
Within a transition metal group moving down the series corresponds with an increase in Δ. The observed result is larger Δ splitting for complexes in octahedral geometries based around transition metal centers of the second or third row, periods 5 and 6 respectively. This Δ splitting is generally large enough that these complexes do not exist as high-spin state. This is true even when the metal center is coordinated to weak field ligands. It is only octahedral coordination complexes which are centered on first row transition metals that fluctuate between high and low-spin states.
The charge of the metal center plays a role in the ligand field and the Δ splitting. The higher the oxidation state of the metal, the stronger the ligand field that is created. In the event that there are two metals with the same d electron configuration, the one with the higher oxidation state is more likely to be low spin than the one with the lower oxidation state. For example, Fe2+ and Co3+ are both d6; however, the higher charge of Co3+ creates a stronger ligand field than Fe2+. All other things being equal, Fe2+ is more likely to be high spin than Co3+.
Ligands also affect the magnitude of Δ splitting of the "d" orbitals according to their field strength as described by the spectrochemical series. Strong-field ligands, such as CN− and CO, increase the Δ splitting and are more likely to be low-spin. Weak-field ligands, such as I− and Br− cause a smaller Δ splitting and are more likely to be high-spin.
Tetrahedral complexes.
The Δ splitting energy for tetrahedral metal complexes (four ligands), Δtet is smaller than that for an octahedral complex. It is unknown to have a Δtet sufficient to overcome the spin pairing energy. Tetrahedral complexes are always high spin. There are no known ligands powerful enough to produce the strong-field case in a tetrahedral complex. 
Square planar complexes.
Most spin-state transitions are between the same geometry, namely octahedral. However, in the case of d8 complexes is a shift in geometry between spin states. There is no possible difference between the high and low-spin states in the d8 octahedral complexes. However, d8 complexes are able to shift from paramagnetic tetrahedral geometry to a diamagnetic low-spin square planar geometry.
Ligand field theory vs Crystal field theory.
The rationale for why the spin states exist according to ligand field theory is essentially the same as the crystal field theory explanation. However the explanation of why the orbitals split is different accordingly with each model and requires translation.
High-spin and low-spin systems.
The first d electron count (special version of electron configuration) with the possibility of holding a high spin or low spin state is octahedral d4 since it has more than the 3 electrons to fill the non-bonding d orbitals according to ligand field theory or the stabilized d orbitals according to crystal field splitting.
Ionic radii.
The spin state of the complex also affects an atom's ionic radius.
d4