Now Paul is starting a comparison between Adam and Christ to show the power of Christ. Paul talked about how from one man sin entered the world, lading to death "because all sinned" (5:12). This is referencing the events of "The Fall" that marked the start of Adam (and Eve) getting separated from God. With the start of sin means the start of death for humans (since sin separates us from God the life provider).
Although sin is in the world, it is technically not held against when there is no law (5:13). But even without legal stuff, sin's effects, like death and ruined relationships, are still there. Paul even talked about how "death reigned [...] even over those who had not sinned according to the likeness of the transgression of Adam" (5:14). Not everyone does exactly the same sin, but everyone ultimately faces side effects of sin like death (since everyone has sinned at least once).
The discussion of this passage raises questions on "original sin" (sin nature from birth, possibly inherited from Adam). Some theologians think so, while other theologians do not and consider doing sin (and fostering sin nature) as human choice. I personally fall into the latter camp (since I firmly believe God created us for good and how we are responsible for our own sin). But either way, we know that everyone has either not chose God or have done some not-so-good things (resulting in side effects like death) and that Adam's (and Eve's) separation from God marked the start of death for humans (because separation from God the life provider). But more importantly, we know that Adam "is a type of Him who was to come". Jesus is similar to Adam in human composition, but Jesus paved a way that counters the effects that started since Adam.