Let $0 \in U$. Define the distribution $\delta \in D'(U)$ as
$$\delta: \phi \rightarrow \phi(0)$$
Show that this really is a distribution. I'm struggling with the definition of a distribution.
Do I need to show that $\delta$ is a linear and continuous map and if so, how could I accomplish that in this case? There is another theorem that states that every integrable function can be seen as a distribution?
Yes, you should show that $\delta$ is a linear and continuous map. Linearity should be close to trivial.
For continuity one has that linear functional $u$ on $C^\infty_c$ is continuous, and thus a distribution, if for each compact set $K$ there exists a number $C_K>0$ and an integer $N_K\geq 0,$ such that for every test function $\phi$ with support on $K,$ one has $$ |\langle \delta, \phi \rangle| \leq C_K \sum_{n\leq N_K} \|\phi^{(n)}\|_K, $$ where $\|\psi\|_K = \sup_{x\in K} |\psi(x)|.$
Hint: For $\delta$ you can take $N_K=0$ and $C_K=1,$ so you only need to show that $|\langle \delta, \phi \rangle| \leq \|\phi\|.$