An orbital is a one-electron wavefunction, usually derived by solving the Schrodinger equation. This tag applies to questions about all forms of orbitals; additionally, questions about the construction and properties of molecular orbitals should be tagged with [molecular-orbital-theory].
An orbital is a one-electron wavefunction, which describes the state of an electron in an atom or molecule.
The form of the familiar atomic orbitals is derived by solving the time-independent Schrodinger equation for the hydrogen atom; this leads to solutions with three quantum numbers ($n$, $l$, and $m_l$).
$n$ is the principal quantum number, and is generally associated with the idea of an electron shell; one shell comprises electrons with the same value of $n$. The allowed values of $n$ are the positive integers.
The azimuthal quantum number, $l$, describes the angular momentum of an electron in such an orbital; s-orbitals have $l = 0$, for p-orbitals $l = 1$, and so on. Therefore, orbitals that share the same values of $n$ and $l$ form a subshell. $l$ is allowed to take integer values between 0 and $(n - 1)$ inclusive; therefore, for the first shell, $(n - 1) = 0$ and only s-orbitals are allowed.
Lastly, the magnetic quantum number, $m_l$, conventionally dictates the projection of the angular momentum onto the $z$-axis. $m_l$ can take integer values between $-l$ and $l$ inclusive; therefore, for a set of p-orbitals with $l = 1$, there are three different orbitals with $m_l=-1,0,+1$.
Further, the fourth quantum number, $m_s$, describes the spin of the electron. It can take two different values, +1/2 and -1/2, and these are associated with "spin up" and "spin down" states respectively. Spin is not accounted for in the Schrodinger equation; in non-relativistic quantum mechanics the existence of spin is a postulate, and it can only be derived from first principles using the relativistic Dirac equation.
Neglecting finer corrections (such as the Lamb shift), the energy of an orbital in a hydrogen-like atom is determined solely by $n$, and is proportional to $-1/n^2$.
In many-electron atoms, the total electronic wavefunction is assumed to be an antisymmetrized product of one-electron wavefunctions (the orbital approximation). This allows the idea of an electronic configuration, where each electron is assigned to an orbital. In these species, the $l$-degeneracy is removed, and in general the orbital energies increase in the sequence: $\pu{1s < 2s < 2p < 3s < 3p < 4s < 3d}\ldots$ According to the Pauli exclusion principle, no two electrons can have the same four quantum numbers. Therefore, each spatial orbital, described by the first three quantum numbers, can only accommodate two electrons, one with spin up and one with spin down. The electronic configuration is then obtained via the Aufbau principle.
Orbitals are not always obtained by solution of the Schrodinger equation for a single electron. So-called "hybrid orbitals", such as sp3 orbitals, are obtained via a linear combination of the atomic orbitals. In all molecules (apart from $\ce{H2+}$), the Schrodinger equation cannot be solved analytically, so molecular orbitals are generated in different ways, most commonly via linear combination of atomic orbitals. In the Hartree-Fock method, the orbitals are obtained iteratively from the series of one-electron Fock equations. In density functional theory, the electronic wavefunction is not the central quantity, but it still proves useful to model the electron density using the fictitious Kohn-Sham orbitals.
