Of course, it depends on how one defines 'person'. If one defines person as an organism with a human genome, then dolphins can't be persons and human fetuses are persons. But personally, I think persons are conscious creatures that are able to think about their own and others' mental states, to represent and understand what they feel and believe, hold dear and hold true. Such self-awareness seems to allow one to represent oneself as a person, as separate from other persons, as continuous through time, as having a future self with interests that should be considered now. These capacities, I believe, are also what make persons autonomous and responsible in ways that non-persons are not. While there are important boundary conditions for personhood, so defined, it may involve many capacities, each of which is possessed to varying degrees, so it may be hard to delineate clearly which creatures count as persons and which don't, and to delineate exactly when an infant becomes a person. (I also don't know whether language is required for any of these capacities or is instead enabled by them.)

So, dolphins are persons if they are conscious (surely they are) and also self-aware in these ways. Some evidence suggests that dolphins, like apes (but not monkeys), have the ability to recognize themselves in a mirror and to represent mental states ('theory of mind'), not to mention their basic language skills. These tests seem to be reasonable ways to get at the capacities I'm suggesting are the grounds for personhood.