The Dirac delta $\delta(x)$ is not an ordinary function. It is a distribution defined by how it acts inside integrals: $$\int_{-\infty}^{\infty}\delta(x-a)f(x),dx=f(a)$$ for suitable test functions $f$. This is the sifting property.

It is useful to think heuristically of $\delta(x-a)$ as concentrated at $x=a$, zero away from that point, with total integral $$\int_{-\infty}^{\infty}\delta(x-a),dx=1.$$ But no ordinary finite-valued function has all these properties simultaneously.

The delta represents idealized point sources. A point mass $m$ at $x=a$ has mass density $$\rho(x)=m\delta(x-a),$$ so $$\int \rho(x),dx=m.$$ In three dimensions, a point charge $q$ at $\mathbf r_0$ is represented by $$\rho(\mathbf r)=q\delta^{(3)}(\mathbf r-\mathbf r_0).$$

Under a change of variable, $$\delta(ax)=\frac{1}{|a|}\delta(x).$$ More generally, for isolated simple roots $x_i$ of $g(x)$, $$\delta(g(x))=\sum_i\frac{\delta(x-x_i)}{|g'(x_i)|}.$$

The delta distribution also expresses orthogonality of continuous eigenstates, impulse forcing and Green-function equations. It is a compact mathematical language for localization, but equations containing deltas are interpreted through integration or distribution theory rather than pointwise arithmetic.