What is the most advanced AI software humans have made to date and what does it do?
In my opinion, this would be Phaeaco, which was developed by Harry Foundalis at Douglas Hofstadter's CRCC research group.
It takes noisy photographic images of Bongard problems as input and (using a variant of Hofstadter's 'Fluid Concepts' architecture) successfully deduces the required rule in many cases.
Hofstadter has described the related success of CopyCat as being 'like a little kid doing a somersault': i.e. it doesn't have the flashy appeal of systems like AlphaGo. What it does however have is a much more flexible (i.e. not precanned) approach to perception of problem structure than other systems, which Hofstadter claims (and many including Peter Norvig agree) is the really hard problem.
In addition to the answers already posted, I think IBM's Watson deserves a mention. It did something pretty impressive with its Jeopardy win, possibly as impressive as AlphaGo. Sadly, since then, there don't seem to have been a lot of really public demos of Watson, as IBM is positioning the technology as a tool for companies and other organizations, and most of them are pretty secretive about the details of what they're doing. I think they did publicize a bit of information about using it for medical diagnosis, but that's the only other application I can think of off hand. I'm sure there are more though.
AlphaGo is the most sophisticated and closet human creation towards an Artificial General Intelligence (AGI). It is a computer program that is developed by Google DeepMind to play the board game "Go". The game is different than other games, as The number of potential legal board positions is greater than the number of atoms in the universe. It has way more legal board positions than the chess. So, AlphaGo requires different technique for it's development.
Program's victories against the best players in the world in March 2016 is considered a major break through in the field of AI. Go was previously considered to be a hard problem and many experts believed that current technology is not enough. Experts were saying that it will take atleast 5 years (or may be 10 years) before we will have a well developed Go software player.
The game used sophisticated algorithms of deep learning and reinforcement learning in order to learn the game. What makes this game different from other board game (like Chess, Reversi, etc.) is that moves are often based on intuition. If you ask a Chess player why he make a certain move, you will always be hearing an answer where he will explain you how he thought this move can increase in change of winning. Every move uses certain heuristics, strategy and/ or tricks. This is not the case with Go. Some moves are often taken because of intuition. Coding an AI software that can play a game, where intuition is a integral part of the game makes it different from other AIs that we have today.
At present AlphaGo is the closest AI software to Artificial General Intelligence.
In my opinion this would be the Google search engine.
It searches the web.