Question: I am wondering about the differences. Based on my understanding, MLP is one kind of neural networks, where the activation function is sigmoid, and error term is cross-entropy(logistics) error. Looking for help, thanks!

Answer: 
You are right, MLP is one kind of neural network.

**There are several kinds of NN**, you can have a NN based on Radial Basis Function with a Soft gating strategy, for example. You can use a committee machine strategy to form a NN...
MLP is fully connected feed-forward network. In particular CNN which is partially connected, RNN which has feedback loop are not MLPs.