What exactly is an orbital?
Terminological questions in chemistry usually are better answered using IUPAC recommended definitions and the IUPAC Gold Book defines orbital (atomic or molecular) 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,
$$
    \left| \psi \right&gt;
    \longleftrightarrow
    \begin{pmatrix}
        \phi_{\alpha}(\vec{r})  \\
        \phi_{\beta}(\vec{r})  
    \end{pmatrix} \, .
$$
This is to be contrasted with a four-component Dirac spinor that describes a state of an individual electron in a relativistic theory,
$$
    \left| \psi \right&gt;
    \longleftrightarrow
    \begin{pmatrix}
        \phi_{\alpha}(\vec{r})  \\
        \phi_{\beta}(\vec{r})   \\
        \chi_{\alpha}(\vec{r})  \\
        \chi_{\beta}(\vec{r})
    \end{pmatrix} \, ,
$$
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,
$$
  \psi(\vec{x}) = \phi(\vec{r}) \alpha(m_s), \quad \psi(\vec{x}) = \phi(\vec{r}) \beta(m_s) \, .
$$