a distribution
A distribution is the rigorous version of a generalized function. The idea is to define an object not by its values at points but by the single number it returns when paired against each smooth test function. Think of a measuring instrument: you never see the thing itself, only the reading it gives against every probe you push in. A distribution is exactly such an instrument, and two distributions are the same precisely when they give the same reading against every probe.
Precisely, a distribution is a continuous linear functional on the space of test functions (the smooth functions that vanish outside a bounded region). Linear means it respects addition and scaling of probes: pairing with phi + psi gives the sum of the two readings, and pairing with c times phi scales the reading by c. Continuous means that if a sequence of probes settles down nicely (their values and all their derivatives converge, all supported in one fixed bounded set), the readings converge too. The pairing of a distribution T with a probe phi is written as the bracket (T, phi). An ordinary locally integrable function f becomes a distribution through the rule (f, phi) = integral of f times phi, so functions sit inside distributions as a special case.
Distributions matter because they are the natural setting for solving PDEs: every distribution can be differentiated infinitely often, point sources and jumps are allowed, and the awkward objects produced by transform methods and fundamental solutions all live here comfortably. The price is one real restriction — you cannot, in general, multiply two distributions — so the theory is built carefully around what is and is not allowed.
The Dirac delta is the distribution whose reading against any probe phi is just (delta, phi) = phi(0). It is linear (it reads phi + psi as phi(0) + psi(0)) and continuous, but there is no ordinary function f with integral of f times phi equal to phi(0) for every probe — so delta is a genuine distribution that is not a function.
A distribution is recognized entirely by the linear, continuous reading it assigns to every test function.
A common confusion: a distribution has no well-defined value at a single point in general (the delta has none at 0). It only makes sense paired against probes, by an integral-like bracket — never evaluated pointwise.