What exactly is an orbital?

What exactly is an orbital? Atomic or molecular.


Is it the function that describes the behaviour of the electron?


Is it the Schroedinger's equation solution, e.g., for Hydrogen atom?


Is it the probability of finding any electron of an atom in any specific region around the atom's nucleus?


Maybe all these describe the same thing. ***What is it?***




Terminological questions in chemistry usually are better answered using IUPAC recommended definitions and the IUPAC Gold Book defines [orbital (atomic or molecular)](http://goldbook.iupac.org/O04317.html) as



> 
> Wavefunction depending explicitly on the spatial coordinates of only
>  one electron.
> 
> 
> 


But to be honest, I don't like this particular definition. The problem is that a one-electron wave function is, strictly speaking, a function of not only the spatial coordinates of an electron ($\vec{r}$) but also the spin one ($m\_s$). In quantum chemistry we distinguish the so-called *spin orbitals*, which are one-electron wave functions, from the so-called *spatial orbitals*, which can be though of wave functions that describes a hypothetical spin-less electron. 


The connection between spin and spatial orbitals is not so trivial. In the most general case a spin orbital $\psi$ can be represented as follows,
$$
 \psi(\vec{x}) = \phi\_{\alpha}(\vec{r}) \alpha(m\_s) + \phi\_{\beta}(\vec{r}) \beta(m\_s) \, ,
$$
where $\vec{x} = \{ \vec{r}, m\_s \}$ is the joint spin-spatial coordinate of an electron, and $\phi\_{\alpha}$ and $\phi\_{\beta}$ are two different spatial orbitals. What we have here is essentially the linear combination of two possible spin states of an electron ("spin-up" and "spin-down") each with its own spatial probability. 


And since OP mentioned in comments the fully relativistic Dirac's treatment of a single electron, it is a good idea to note an alternative way of expressing the idea of a spin orbital: in a form of the so-called *two-component wave function*, or *Pauli spinor*, that represents the state of a single electron in a non-relativistic theory,
$$
 \left| \psi \right>
 \longleftrightarrow
 \begin{pmatrix}
 \phi\_{\alpha}(\vec{r}) \\
 \phi\_{\beta}(\vec{r}) 
 \end{pmatrix} \, .
$$
This is to be contrasted with a four-component *Dirac spinor* that describes a state of an individual electron in a relativistic theory,
$$
 \left| \psi \right>
 \longleftrightarrow
 \begin{pmatrix}
 \phi\_{\alpha}(\vec{r}) \\
 \phi\_{\beta}(\vec{r}) \\
 \chi\_{\alpha}(\vec{r}) \\
 \chi\_{\beta}(\vec{r})
 \end{pmatrix} \, ,
$$
in which the last two components describe a "spin-up" and a "spin-down" states of the associated positron.


Now, back to chemistry, we note that in general chemistry we always think of each and every electron as being in a pure spin sate: either "spin-up" or "spin-down", rather than in a superposition of these two. Besides, we assume that spin orbitals come in pairs: for each and every spatial orbital $\phi(\vec{r})$ we have two spin orbitals given as follows,
$$
 \psi(\vec{x}) = \phi(\vec{r}) \alpha(m\_s), \quad \psi(\vec{x}) = \phi(\vec{r}) \beta(m\_s) \, .
$$


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I decided add this answer because I have a different viewpoint, although I agree with Wildcat on the IUPAC definition.


*What exactly is an orbital? Atomic or molecular.*


The term *orbital* is used with more than one meaning. Normally they are classified into different kinds. In a technical sense, they are *just* functions of the variables on only one electron. If the variables are spatial coordinates they are called *spatial orbital*, if they also include the spin coordinate they are called *spin orbitals*.
Generally speaking (for more than one electron particles) they are just mathematical constructions useful get the wavefunction or an approximation to it (from which we can get estimation of measurable values).


*Is it the Schroedinger's equation solution, e.g., for Hydrogen atom?*


I assume that you are speaking about the Time Independent Schrödinger Equation (TISE). In such case, generally not. In the particular case of H atom they are solutions of TISE, as well as linear combinations of them. In such case they represent stationary states of the system, in other words, are functions that represent (somehow) time independent states in which the system can be in (but only in systems with one electron!).


In polyelectronic species, they represent nothing, although it can be given (in some context) some approximate interpretations. Going further, if you decompose the system into one particle systems, and get their orbitals, it is mathematically true (because this: [see last example](https://en.wikipedia.org/wiki/Tensor_product_of_Hilbert_spaces#Examples_and_applications)) that they can be combined (how to is the hard part) to represent the wavefunction of the original system.


*Is it the probability of finding any electron of an atom in any specific region around the atom's nucleus?*


No. They are rather arbitrary. It is a common practice to transform them according to one needs. In any case, normally they represent something like the stationary state the one electron embed in a space region where it feels the mean interaction with the remaining electrons. But do not dig to much in those interpretations. 


*Maybe all these describe the same thing. **What is it?***


The introductory textbooks call orbital to an space region. It is not compatible with the usage of the word above. In a polyelectronic system, sometimes, for some kind of orbitals, in very well defined contexts, they can have a sort of interpretation, in essence they are just helpful functions.


You should check the answers to this question: [What does an orbital mean in atoms with multiple electrons?...](https://physics.stackexchange.com/questions/224108/what-does-an-orbital-mean-in-atoms-with-multiple-electrons-what-do-the-orbitals) They give some details that I omitted.


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