Neural networks types include perceptrons, Hopfield networks, Boltzmann machines, fully connected neural networks, convolutional neural networks, re-current neural networks, long/short term memory neural networks, auto-encoders, deep belief networks, generative adversarial networks. Majority of them are trained with an algorithm called back-propagation.
Convolutional neural networks which could be considered essentially a not fully connected neural nets in which each neuron is connected to only a few neurons in the previous layer and neurons share weights. These type of networks have been proven successfully especially in the fields of computer vision and natural language processing, where they broke every record.