I just noticed the question. The references mentioned so far are good. So I'll just do the example that I normally do on a blackboard when anyone asks me. Take a smooth complex projective variety $X$ with a smooth divisor $D$. Set $U=X-D$. There's a long exact sequence $$ \ldots H^i(X)\to H^i(U)\to H^{i-1}(D)\to H^{i+1}(X)\ldots$$ say with rational or complex coefficients. What are the maps? The first is restriction, the second using $\mathbb{C}$ coefficients is a residue map, and the third is the Gysin map which is of type $(1,1)$ (or you want to want to get fancy you need a Tate twist here). The mixed Hodge theory of $U$ can be read off from this.
For example, for the Hodge numbers $$\dim H^{pq}(U)= \dim im[H^{p-1,q-1}(D)\to H^{pq}(X)]+\dim ker[H^{pq}(D)\to H^{p+1,q+1}(X)]$$ and so on. (By the way, $H^{pq}$ is taken to be the $(p,q)$ part of the $p+q$ weight graded quotient.)
OK, let me make it more concrete. Let $X$ be a surface with irregularity $q=0$, perhaps $\mathbb{P}^2$, then $D \subset X$ is a curve of say genus $g$. Then from above, the interesting Hodge numbers are $$h^{20}(U)=h^{02}(U)=h^{20}(X)$$ $$h^{11}(U)=h^{11}(X)-1$$ $$h^{12}(U)=h^{21}(U)=g$$
Maybe that's enough for now.
I can't resist squeezing in one more example. Suppose $D$ on the above surface $X$ can be contracted to a point in a normal surface $Y$. So for example, $Y$ might be a cone over a plane curve, and $X$ the blow up of the vertex. Using duality and the standard exact sequence for compactly supported cohomology, we can conclude $$H^i(Y)=H_c^i(U)= H^{4-i}(U)^*(-2)$$ H^{4-i}(U)^*(-2),\quad i>0$$ As far as Hodge numbers are concerned, the dual means $(p,q)\mapsto (-p,-q)$ and $(-2)$ means shift by $(2,2)$.
