atomic orbital
An atomic orbital is the wavefunction of a single electron in an atom — a mathematical object that, when you take the square of its size, tells you the probability of finding that electron at each point around the nucleus. The familiar pictures of spheres, dumbbells, and four-leaf-clover shapes are simply maps of where that probability piles up. Each orbital is fixed by a set of quantum numbers that pin down its energy, its shape, and its orientation.
The word 'orbital' is chosen carefully to break with the older word 'orbit'. An electron does not trace a path the way a planet circles the Sun; it has no definite trajectory at all. What an orbital describes is a standing pattern of probability, smeared out in three dimensions and unchanging in time when the electron is in a steady state. Asking where the electron is right now, between observations, is a question the theory declines to answer.
Orbitals are the building blocks of all of chemistry. By learning which orbitals exist and how many electrons each can hold, we can explain why atoms bond as they do, why the periodic table has the columns it has, and why some substances react eagerly while others sit inert. The exact orbitals are solved only for hydrogen, but the same shapes, suitably adjusted, serve as the working vocabulary for every other atom.
An orbital is a one-electron wavefunction; its squared size is where the electron is likely to be.
The colourful orbital shapes are surfaces enclosing most of the probability, not solid objects. The electron cloud has no sharp edge — it merely fades smoothly to near zero far from the nucleus.