The Dirac delta “function” $\delta(x)$, or distribution is not a true function because it is not uniquely defined for all values of the argument $x$. Similar to the Kronecker delta symbol, the notation $\delta(x)$ stands for

For any continuous function $F$:

or in $n$ dimensions:

$\delta(x)$ can also be defined as a normalized Gaussian function (normal distribution) in the limit of zero width.

Notes: However, the limit of the normalized Gaussian function is still meaningless as a function, but some people still write such a limit as being equal to the Dirac distribution considered above in the first paragraph.
An example of how the Dirac distribution arises in a physical, classical context is available on line.

Remarks: Distributions play important roles in Dirac’s formulation of quantum mechanics.