This paper provides full classification of dynamics for continuous time Markov chains (CTMCs) on the non-negative integers with polynomial transition rate functions. Such stochastic processes are abundant in applications, in particular in biology. More precisely, for CTMCs of bounded jumps, we provide necessary and sufficient conditions in terms of calculable parameters for explosivity, recurrence vs transience, certain absorption, positive recurrence vs null recurrence, and implosivity. Simple sufficient conditions for exponential ergodicity of stationary distributions and quasi-stationary distributions as well as existence and non-existence of moments of hitting times are also obtained. Similar simple sufficient conditions for the aforementioned dynamics together with their opposite dynamics are established for CTMCs with unbounded jumps. The results generalize respective criteria for birth-death processes by Karlin and McGregor in the 1960s. Finally, we apply our results to stochastic reaction networks, an extended class of branching processes, a general bursty single-cell stochastic gene expression model, and population processes, none of which are birth-death processes. The approach is based on a mixture of Lyapunov-Foster type results, semimartingale approach, as well as estimates of stationary measures.