There are many, some are 'problems' and some are 'techniques'.

    Automatic Programming - The task of describing what a program
        should do and having the AI system 'write' the program.

    Bayesian Networks - A technique of structuring and inferencing
        with probabilistic information.  (Part of the "machine learning"
	problem).

    Constraint Statisfaction - solving NP-complete problems, using a
        variety of techniques.

    Knowledge Engineering/Representation - turning what we know about
	a particular domain into a form in which a computer can
	understand it. 

    Machine Learning - Programs that learn from experience or data.

    Natural Language Processing(NLP) - Processing and (perhaps)
        understanding human ("natural") language.  Also known as
	computational linguistics.

    Neural Networks(NN) - The study of programs that function in a
        manner similar to how animal brains do.

    Planning - given a set of actions, a goal state, and a present state,
	decide which actions must be taken so that the present state
	is turned into the goal state

    Robotics - The intersection of AI and robotics, this field tries
        to get (usually mobile) robots to act intelligently.

    Speech Recogntion - Conversion of speech into text.

    Search - The finding of a path from a start state to a goal
        state. Similar to planning, yet different...

    Visual Pattern Recognition - The ability to reproduce the
	human sense of sight on a machine.

AI problems (speech recognition, NLP, vision, automatic programming,
knowledge representation, etc.) can be paired with techniques (NN,
search, Bayesian nets, production systems, etc.)  to make distinctions
such as search-based NLP vs. NN NLP vs. Statistical/Probabilistic NLP.
Then you can combine techniques, such as using neural networks to
guide search.  And you can combine problems, such as posing that
knowledge representation and language are equivalent.  (Or you can
combine AI with problems from other domains.)