GOFAI (Good Old Fashioned Artificial Intelligence) is a term coined by John Haugeland in
Artificial Intelligence: The Very Idea (1985). In simple terms, this refers to an artificial intelligence that works by taking in a rule set and outputting working solutions.
This includes programs that try to brute-force calculate the optimal moves in games of checkers and chess and the expert systems of the 1980s. These systems were very 'brittle', meaning that even a small error in input could derail them, and they learned by being given new rules or facts, rather than deducing new rules or facts. This sort of AI is fairly weak by today's standards, but it is important to note that GOFAI is what most people think of when they think of an AI, e.g. Deep Blue and ELIZA.
Modern AI systems are enhanced by artificial neural networks, evolutionary computation, and deep learning, among other techniques.