Do only animals have a nervous system?
Plants don't have central nervous systems for the same reason fungi, microbes, and sea sponges don't: they didn't evolve in those groups, and didn't need to. Plants don't need brains any more than humans need leaves, and it wouldn't help them (Shelomi, 2012).
However, they can transmit information from parts to parts in a very similar way to our own nervous system. They can "communicate" within their body via changes in water pressure and certain compounds like hormones, and can "communicate" with other plants in a limited way with chemicals too.