Is it well known what happens if one blowsup $\mathbb{P}^2$ at points in nongeneral position (ie. 3 points on a line, 6 on a conic etc)? Are these objects isomorphic to something nice?
I'll just add to Francesco's answer by saying that general position of the points on the plane is equivalent to ampleness of the anticanonical sheaf $\omega_X^{\otimes 1}$. The key observation is that on a del Pezzo surface, an irreducible negative curve ($C^2 < 0$) must be an exceptional curve (i.e. $C^2 = C\cdot K_X = 1$). This follow from the adjunction formula and the NakaiMoishezon criterion. If you blow up 3 colinear points, then the strict transform of the line containing these points will have selfintersection $2$, which is not allowed by the key observation. Similiarly for the strict transform of a conic through 6 blownup points. There is one more condition you have to impose: if you blow up 8 points, they cannot lie on a singular cubic with one of the points at the singularity. If you relax the requirement that $\omega_X^{\otimes 1}$ be ample to just big and nef, then you can have some degenerate point configurations: this time $C^2 = 2$ is allowed, so 3 colinear points ar OK. However, 4 colinear points would not be OK. 


In both examples you are considering, the anticanonical model is a singular del Pezzo surface. In fact, let $X$ be the blowup of $\mathbb{P}^2$ at three points lying on a line $L$. By Bezout's theorem, the birational map associated with the linear system of cubics through the three points contracts $L$. Since $L^2=1$, the blowup of $L$ at three points gives a $(2)$curve. Therefore, the anticanonical model of $X$ is a Del Pezzo of degree $6$ in $\mathbb{P}^6$ with an ordinary double point (i.e., a node). Analogously, let $Y$ be the blowup of $\mathbb{P}^2$ at six points lying on a conic $C$. By Bezout's theorem, the birational map associated with the linear system of cubics through the six points contracts $C$. Since $C^2=4$ and we are blowingup six points over $C$, we obtain again a $(2)$curve. Therefore, the anticanonical model of $Y$ is a Del Pezzo surface of degree $3$ in $\mathbb{P}^3$ with an ordinary double point, i.e. a cubic surface with a node. Of course, if you blowup more than $8$ points then the result is not a Del Pezzo anymore. For instance, the blowup of $\mathbb{P}^2$ at nine points which are the base locus of a pencil of cubics is an elliptic fibration $X \to \mathbb{P}^1$ with nine sections; in general, such fibration has exactly $12$ nodal fibres, corresponding to the singular elements of the pencil. When the number of points increases the situation becomes more and more complicated, and I guess that a satisfatory description is out of reach. 

