They are in some sense the opposite of surfaces of general type, whose canonical class is big.
When the degree is 2, we have to add the condition that the point is not fixed by the Geiser involution, associated to the anti-canonical morphism.
Conversely any blowup of the plane in points satisfying these conditions is a del Pezzo surface.
The Picard group of a del Pezzo surface of degree d is the odd unimodular lattice I1,9−d, except when the surface is a product of 2 lines when the Picard group is the even unimodular lattice II1,1.When it is an odd lattice, the canonical element is (3, 1, 1, 1, ....), and the exceptional curves are represented by permutations of all but the first coordinate of the following vectors: Degree 1: they have 240 (−1)-curves corresponding to the roots of an E8 root system.
This map is generically 2 to 1, so this surface is sometimes called a del Pezzo double plane.
The 56 lines of the del Pezzo surface map in pairs to the 28 bitangents of a quartic.
They have 27 (−1)-curves corresponding to the minuscule vectors of one coset in the dual of the E6 lattice, which map to the 27 lines of the cubic surface.
The blowup of any point on a weak del Pezzo surface is a weak del Pezzo surface of degree 1 less, provided that the point does not lie on a −2-curve and the degree is greater than 1.
Any curve on a weak del Pezzo surface has self intersection number at least −2.