What is the origin of the Titans in Attack on Titan?

There are a few explanations for the origin, but it all starts with a founding titan called Yimir Fritz.

2000 years ago, a girl named Ymr encountered a devil of the earth. This devil empowered her to become a Titan at will, and gave her power to control over her subject (Eldians).

Then, Ymir came into contact with pure matter which granted her with the Titan ability. Grisha, who was known as an Eldian Restorationist, thought that Ymir was a benevolent goddess and made the earth prosperous.

Now, the story has a twist.

It has been theorized that about 4000 years ago, a group of people alongside Ymir Fritz became scientists and conducted several experiments related to eugenics. The experiments were intended to create a new generation of humans who are stronger, faster, and bigger. As a result, a new human specie was created. The race of the Eldians, who were under the control of Ymir, was that. Eldians could turn into titans at will. In ancient times, the Eldians used titans to build their empire, slaughtering countless peoples and taking their land.

The race of the Eldians

It was believed that when Ymir died, all these titans borne out of the genetic experiments (including the Colossal, the Armour, The Attack, The Founding, the Jaws, The Beast, The Female, The Cart and the Warhammer) were divided into nine pieces. It was also believed that the one who possess the Founding titan, also known as the Coordinate, could control Eldians.

The Founding titan, also known as the Coordinate

The Ackerman clan, another titan came out of these experiments, is the one to focus here. They were designed to protect their hosts, namely the royal family such as the Fritz or the Reiss. Thus, they were a bloodline of warriors. Surprisingly, the Coordinate power did not affect them, because they possessed the non-Eldian blood. Perceiving this as a potential threat, the royal family finally persecuted most of the Ackermans. Many of them died and only two living members were alive at the start of the anime. 

In Attack on Titan ANSWERS, Hajime Isayama stated that "[Mikasa, Levi, and Kenny] are all part of the same Ackerman bloodline."

This is how the story of Attack on Titan begins.