A Dirac distribution or Dirac $\delta$-distribution $\delta(p)$ is the distribution that is given by evaluating a function at a point $p$.
That is, the Dirac $\delta(p)$ function is the distribution defined by $$\langle\delta(p),\phi \rangle=\phi(p)$$
Suppose we want to write
$$\int\phi(x)\phi(x)dx=\int\int_0^1\delta( x-y)\phi(x)\phi(y)dxdy$$
How should we define $\delta( x-y)$?
In this page Free quantum fields on example 14.4 they have
$\delta(
x-y) \in \Gamma'(E \boxtimes E)$ where $\Gamma'(E \boxtimes E)$ is the dual of the space of section of the bundle $\Gamma(E \boxtimes E)$ and
And $E$ is the real line bundle
Define $\delta(x-y)$ for $x,y\in\mathbb{R}^n$ as the distribution on $\mathbb{R}^n\times\mathbb{R}^n$ such that $$ \langle \delta(x-y), \varphi(x, y) \rangle = \int \varphi(t, t) \, dt $$ for every test function $\varphi.$