Weak AI: Artificial intelligences that focus on one narrow task. Most existing products that claim themselves to be "AI" are weak AIs. Both Siri and a calculator belongs to this category.
Strong AI / Artificial General Intelligence: Artificial intelligences that can solve any intelligent task a human can. Many are researching this field, but as of today it's only existed in science fiction.
AGI research also looks into Artificial Consciousness, since that's an important of what makes us human. Some define Strong AI as machines that has consciousness.
Machine Learning: The technique of training a machine to improve performance of some specific task. Machine learning learns how to better perform a task solely by looking at examples of that task (training data).
Neural Network: A computing system vaguely inspired by the network of neurons in our brain. Neural network was better than previous ML frameworks on some fields, such as image recognition & game playing.