I sort of understand the intuition about the definition that if you have for example a constant function the you approximate the integral of the constant function with a first order polynomial, then the errors cancel out, therefore it has a degree of exactness one. But it would help me a lot if someone had an example which they could share so I can understand it better. Thank you!
Definition: Let $[a,b]$ in $\Bbb R$ with $a < b$, and consider a quadrature formula: $$ Q[f;a,b] = \sum_{k=0}^n w_k f(x_k) $$ given in the terms of nodes $x_0 ,... ,x_n$ in $[a,b]$ with $x_0 < ... < x_n$ and weights $w_0, ... ,w_n$ in R. Then $Q$ is called exact of degree $m$ (in the natural numbers) if and only if for every basis $\{ p_0 , ..., p_m\}$ of $P_m$, we have: $$ Q[p_j;a,b] = \int_a^b p_j(x) dx,~~~ j=0,....,m $$
I think that the key step is to recognize that it is enough to investigate a single basis for the space of polynomials of degree at most $m$, rather than every basis.
Specifically, if $\{p_j\}_{j=0}^m$ is a basis for $\mathbb{P}_m$ and $$ \forall j \in \{0,1,2,\dotsc,m\} \: : \: Q(p) = \int_a^b p_j(x)dx,$$ then we automatically have $$ \forall p \in \mathbb{P}_m \: : \: Q(p) = \int_a^b p(x)dx = I(p).$$ This follows from the linearity of $Q$, i.e., if $$p(x) = \sum_{j=0}^m r_j p_j(x),$$ then $$ Q(p) = Q\left(\sum_{j=0}^m r_j p_j\right) = \sum_{j=0}^m r_j Q(p_j) = \sum_{j=0}^m r_j \int_a^b p_j(x)dx = \int_a^b \sum_{j=0}^m r_j p_j(x)dx = \int_a^b p(x)dx.$$ In short, there is no need to examine more than a single basis.
As a specific example, we consider the simple trapezoidal rule on the interval $I_h = [-h,h]$. Here $Q$ is the simple rule given by $$Q(f) = h (f(-h) + f(h)).$$ We choose $p_j : \mathbb{R} \rightarrow \mathbb{R}$ to be the polynomial given by $p_j(x) = x^j$. Then $$Q(p_0) = 2h, \quad Q(p_1) = 0, \quad Q(p_2) = 2h^3$$ and the integrals are $$I(p_0) = 2h, \quad I(p_1) = 0, \quad I(p_2) = \frac{2}{3} h^3.$$ We see that the trapezoidal rule is exact for all polynomials of degree at most 1. It follows that the degree of exactness is $m=1$.