Let $f,g$ be distributions where one of them has a compact support. Then the convolution $f ∗ g$ is defined as
$\langle f ∗g, \phi \rangle= \langle f,\psi \rangle$ where $\psi= \langle g,\phi(x+\cdot) \rangle$
Could somebody explain to me what the dot $\cdot$ means in the definition?
There is also a definition that states $\langle f∗g,\phi \rangle =\langle f,\phi∗\overline{g} \rangle $ where $\overline{g}(x)=g(-x)$. How to understand that in this context?
$\phi(x+\cdot)$, where $x$ is fixed by some other scope, is the function that maps $y$ to $\phi(x+y)$. In this case $x$ is the argument of $\psi$, which is then "integrated over" in $\langle f,\psi \rangle$.
The intuition behind these definitions are formal calculations, written in terms of integrals against functions even though distributions aren't actually given by integration against functions.
Either way the thing you would like to start with is
$$\int_{-\infty}^\infty \int_{-\infty}^\infty f(x-y) g(y) \phi(x) dy dx.$$
We want to rewrite this as $\int_{-\infty}^\infty f(z) \psi(z) dz$ for some $\psi$. To get the first definition, take $u=x-y,v=y$ so that $x=u+v,y=v$. Then you have
$$\int_{-\infty}^\infty f(u) \int_{-\infty}^\infty g(v) \phi(u+v) dv du$$
which is the idea behind the first definition.
You get the connection to the second definition by looking at $w=-v$, which gives
$$\int_{-\infty}^\infty f(u) \int_{-\infty}^\infty g(-w) \phi(u-w) dw du.$$