A Simpler way of Thinking About Integrability

Question

I have recently been looking into the concept of integrability with regard to dynamical systems, and the more I look into it, the less sense it seems to make. There seems to be several kinds of integrability and their definitions tend to be quite dense and technical. On a very high level, what does it mean for a system to be integrable? Was integrability an old concept which took a lot of work to rigorously define? Are there any text books which give a good intuitive grasp of integrablilty? If it helps I'm working with Hamiltonian dynamical systems.