So called "identical twins," which are formed by an ovum that split into two, generally have the exact same genome. There are a few instances, however, when the genetic material can be modified after the split.
So called "fraternal twins," which are formed by two entirely different ova that just happen to have been released in the same menstrual cycle, have different genomes.
Yes as Chroot said, monozygotic twins have the exact same DNA, although a few instances are known where the karyotype for instance was altered in one of the twins, giving rise to a boy-girl identical twin. Dizygotic twins per definition are independently fertilized zygotes.
Galton's Anecdotal Report on Twins. The use of twins to study the inheritance of behavioral traits was another of Galton's important "firsts." He noted that there were two types of twins, judging from their degree of resemblance. "Identical" twins come from one egg (hence they are now called monozygotic, or MZ, twins), which divides in two shortly after fertilization. Their genetic makeup is identical; thus their genetic correlation is unity (r = 1). And they are very alike in appearance. "Fraternal" twins (now called dizygotic, or DZ) come from two different fertilized eggs and have the same genetic relationship as ordinary siblings, with a genetic correlation of about one-half (on average). That is, DZ twins are, on average, about one-half as similar, genetically, as MZ twins. DZ twins are no more alike in appearance than ordinary siblings when they are compared at the same age.
Galton was interested in twins' similarities and differences, especially in MZ twins, as any difference would reflect only the influence of environment or nongenetic factors. He located some eighty pairs of twins whose close physical resemblance suggested they were MZ, and he collected anecdotal data on their behavioral characteristics from their relatives and friends and from the twins themselves. He concluded that since the twins were so strikingly similar in their traits, compared to ordinary siblings, heredity was the predominant cause of differences in individuals' psychological characteristics. Because Galton obtained no actual measurements, systematic observations, or quantitative data, his conclusions are of course liable to the well-known shortcomings of all anecdotal reports. Later research, however, based on the more precise methods of modern psychometrics and biometrical genetics, has largely substantiated Galton's surmise about the relative importance of heredity and environment for individual differences in general mental ability.