Artificial neural networks (ANNs) are computing systems inspired by the human brain. They can teach themselves to do tasks, simply by considering examples of the tasks’ outcome. For example, they can learn to identify images that contain cats by analyzing example images that have been tagged “cat” or “no cat”. When given enough examples, the neural network can autonomously determine whether “untagged” images include cats or not (Wikipedia). If you want to learn more and have 20 minutes to spare, I can recommend this YouTube video by Brandon Rohrer.
Neural networks are commonly used for those machine learning problems where there is a vast amount of (complex) data available. Some toy examples include fingerprint recognition, language translation, car steering behaviours, object detection, text generation, and doodle recognition (by Google). Chances are pretty high that any system that makes complex recommendations these days (e.g., “Is this John in the picture?”, “Did you mean “South End Taco’s” instead of “Sout En dTacos”?”) has a neural net running in the background.
http://www.r-exercises.com designs tutorials for beginning programmers in R. On their website they host a learning series on neural networks, consisting of three sets of exercises: Part 1, Part 2, and Part 3. Afterwards, you can check your performance with the solutions: Solutions 1, Solutions 2, and Solutions 3.
Keep on learning!
P.S. afterwards you might want to check out this package and API for deep learning in R and Python.