One way is to take advantage of the natural principles of gravity and suction. If you use the right proportions, you can set up a system that flows down from a higher point and back around simply from the pressure of the water falling.
In this case, the higher your source, the stronger the pump and flow will be. When the water flows in and out of your system, it creates a suck that will move the contents of your pipe.
Another way you can pump water without electricity is to use renewable energy. With the current climate crisis, taking advantage of renewables is an excellent step in the right direction.