Convolutional Networks work by moving small filters across the input image. This means the filters are re-used for recognizing patterns throughout the entire input image. This makes the Convolutional Networks much more powerful than Fully-Connected networks with the same number of variables. This in turn makes the Convolutional Networks faster to train.
In this notebook we implemented a simple Convolutional Neural Network in TensorFlow which has a classification accuracy of about 99%.