Most goods can only be consumed by one person, or by one person at a time. Economists call such goods rival because consumption of them is competitive in a sense. A typical rival good might be pizza -- although several people can share a pizza, each individual bite can only be eaten by one person. Many other goods are also rival, although it is sometimes less obvious. A car, for example, can only be driven by one person at a time.
Goods that do not have this property are called nonrival. A typical example might be a national park. Provided people don't litter or tear up the park, one person's use of it doesn't diminish the ability of other people to use it. Of course, with parks this is only true up to a point -- if too many people try to use it at once it becomes congested and is no longer nonrival.
A key aspect of nonrival goods arises when computing the marginal benefit of providing one more unit of the good. With an ordinary rival good like a car, the marginal benefit of one more unit (an extra car) is equal to the extra benefit received by the individual who receives that unit (that particular car). With nonrival goods, however, everyone who uses the good at all can benefit from an additional unit of it. If we enlarge a park, for example, everyone who uses the park can benefit. This means that for nonrival goods, the marginal benefit of providing an extra unit is the sum of the marginal benefits received by each of the individual users.