The word ‘period’ has many meanings in mathematics, most of them coming from physics: the period of an oscillation, periods in celestial mechanics, even the period of a periodic function? comes from the intuition that the periodicity is in the time dimension. Functions on tori are periodic in two directions, say the Weierstrass functions? on elliptic curves, so it is not a surprise that more involved kinds of periods came from the study of elliptic curves and then more general Riemann surfaces.
The period of a closed differential form $\omega \in \Omega^n_{cl}(X)$ over an $n$-cycle $S$ is the integral $\int_S \omega$.
There is another deep notion of periods in number theory and a more specific version related to specific situations in algebraic geometry. We distinguish irrational and rational numbers; complex numbers divide into algebraic and transcendental. Periods are more general than algebraic numbers: they are those (complex) numbers which can be obtained as integrals of algebraic functions (all of whose coefficients are also algebraic numbers) over semialgebraic sets. The periods form a subring of complex numbers bigger than the field of algebraic numbers. There are several operations which lead to new periods. In fact, if we view them abstractly, as integrals of some abstract function over an abstract semialgebraic set, then we can take unions of such sets, do partial integration and so on. There is a conjecture that there are no relations among periods except those of a short list of such obvious relations!
Periods appear in a number of situations in classical algebraic geometry. Specific matrices of periods are defined and important in the theory of algebraic functions, Hodge theory for algebraic cycles, the study of actions of motivic Galois groups, etc. They come as generalizations of “periods of Riemann surfaces” from 19th century.
A general introduction to and discussion of algebraic periods is in
and a popularization in
See also
A. B. Goncharov, Periods and mixed motives, math.AG/0202154
mathoverflow: ring of periods not a field
Discussion of motives in physics via periods as appearing in the deformation quantization of free field theories, hence in correlators/scattering amplitudes, and their relation to the cosmic Galois group originates in
More details on this (and a good review of periods in the first place) is in
and briefly in section 8.5 of
For more see at motives in physics.