AI surely will dominate in the future, it is currently applied to speech recognition, self-driving cars, image processing, etc. and has the potential to be applied in every process our brain is capable of, even more, it has the potential to surpass human intelligence. If that is the case we should invent methodologies to prevent our risk of existence because of the domination AI will have on us, the same domination we have now over other less intelligent beings like ants or dogs or pigs etc. For more information on this particular problem read the Superintelligence: Paths, Dangers, Strategies by Nick Bostrom. 

The main algorithms in Artificial Intelligence are taken from reverse-engineering the brain with a bit of help from mathematics and cognitive science. The first idea of a computer being equally intelligent with a human came with the Turing Test (check his 1950 paper) and the fundamentals of AI came later. Check this 1961 film if you want to see more about past ideas behind thinking and thinking machines.

However, many had passed and technology evolves exponentially fast, from alphabet we came to complex images, from ELIZA to Google Assistant, from checkers to chess, Jeopardy!, Go, Dota. To have a grasp of more modern AI you can read Python Machine Learning, Deep Learning or the behemoth Artificial Intelligence: A Modern Approach. You might also check this blog or other publications, and other.

If you are not much of a reader, check the MIT courses on Deep Learning, open.ai resources, fast.ai, deeplearning.ai, Intro to AI and Intro to ML courses from udacity.com. Or you can see the following videos to get a general idea.

Cool lectures from different topics relating to intelligence delivered by the MIT institute of Brains, Minds and Machines.

AI introductory lectures help by Patrick Winston.