Devil May Cry begins with Dante being attacked in his office by a mysterious woman named Trish. He impresses her by easily brushing off her assault, and explains that for years, he has hunted demons in pursuit of the ones who killed his mother and brother. Trish explains that her attack was a test, and that the demon emperor Mundus, whom Dante holds responsible for the deaths of his family, is planning to cross over into the human world after centuries of imprisonment.
The gameplay consists of levels called "missions", where players must fight numerous enemies, perform platforming tasks, and occasionally solve puzzles to progress through the story. The player's performance in each mission is given a letter grade, starting with D, increasing to C, B, and A, with an additional top grade of S. Grades are based on the time taken to complete the mission, the amount of "red orbs" gathered (the in-game currency obtained from defeated enemies, destroyed objects, and exploration), how "stylish" their combat was, item usage, and damage taken.
Source: Wikipedia, "Devil_May_Cry_(video_game)," available under the CC-BY-SA License.