What is Irrigation?
Often people farm in a place where enough rain falls during the year (and at the right times) to water the plants just with rainfall. Farmers don't have to worry about the plants getting enough rain. That's called "dry farming" because the farmers don't have to carry water to the plants.
But in other places, like Egypt or the Arabian peninsula, it hardly rains at all. Farmers can't rely just on the rainfall to water their crops. They have to find some way of getting water from the river to their fields. That's called "wet farming," and the way they get water from the river is called "irrigation."
Shaduf in Egypt
There are lots of different ways to get water from the river to the fields. One way is just to carry it yourself in buckets, and plenty of people irrigated this way in the ancient and medieval periods (and plenty of people still do today). But it's very hard work carrying water in buckets, and you can't carry very much even if you work very hard. Even a donkey or an ox can't carry enough water to irrigate big fields.
So whenever they can, people use some kind of machine to help them carry the water. Often this is a lever: a long wooden pole with a bucket on one end so people can hold the other end of the pole and lower the bucket into the water and then raise it and swing it around and dump it into a canal that is a little higher up, and that carries the water to the fields. In Egypt, people call this machine a shaduf.
Or sometimes people will use a water wheel to lift the water up and dump it into the canal. That way the water can power the lifting as well as feeding the crops.
Ancient Agriculture: From Foraging to Farming, by Michael and Mary Woods (2000). For middle schoolers, with plenty of information about how farming got started, and how it worked.
Engineering in the Ancient World, by John Landels (revised edition 2000).
Greek and Roman Technology : A Sourcebook, edited by John Humphrey, John Oleson, and Andrew Sherwood (1998). A collection of essays by specialists about different things people made or invented around the ancient Mediterranean, including irrigation machines like water wheels. Humphrey is a very careful and thorough researcher.
Greek and Roman Technology, by K. D. White (1984). The classic in this field for the last twenty years. Pretty easy to understand.