Orbital
Definition
A region around the nucleus where an electron is most likely to be found. Different orbitals have different shapes: spheres, dumbbells, and more complex forms.
Technical definition
An atomic orbital is a one-electron wavefunction describing a region of space with high probability of finding an electron, characterized by the quantum numbers n, l, and ml. S orbitals are spherical, p orbitals are dumbbell-shaped, and each orbital holds at most two electrons with opposite spins.
Example
The 2p orbitals of carbon, three dumbbell-shaped regions at right angles, are where carbon's bonding electrons live.
Common confusion
Orbitals are not fixed paths like planetary orbits. They are probability clouds: you cannot say exactly where the electron is, only where it is likely to be.