Dynamic programming is the very powerful mathematical tool. When applied to economics, we can use that method to solve, well, (almost) all problems (even if the model is stochastic!). The problem is that many books on this method are too technical for economists and it's hard to understand it.
The most accessible introduction to dynamic programming is, in my opinion, George McCandless's "ABCs of RBCs" (his lecture notes are also nice). The second best is Klaus Walde's Applied Intertemporal Optimization (AIO for short, available for free!). McCandless doesn't explain the continuous-time dynamic programming, so if you work with continuous-time models, you should read AIO as well. The final book on dynamic programming is the bible - Recursive Methods in Economic Dynamics. However, this classic is too technical. Unless you are familiar with abstract mathematics, this book cannot be recommended.
By the way, dynamic programming is developed by Richard E. Bellman. That's why you have Bellman equation when solving problems using dynamic programming.
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