I can't say definitively whether or not it's absolutely the oldest, but the oldest in my part of the world (which is a very old part of the world) is the Marib Dam in Yemen which dates to the 8th century BC, is still standing (albeit of course renovated many times over and has failed in the past) and may in fact be older: https://en.wikipedia.org/wiki/Marib_Dam

Edit: to expand on your other questions, my understanding is that it, like most dams in Yemen, was built to control flash flooding and preserve annual rains for irrigation purposes in order to regularize water flows in what is otherwise quite an arid region. These dam systems make Yemen and Oman quite green.

Depends if you count fishing weirs. If yes than at least 8000 years. Which one is the oldest? I'm not sure if anyone keeps track of that since they were fairly common. This document for example gives you the site of of 9 found in Denmark alone on page 2:

https://www.york.ac.uk/archaeology/middens/Chap%2005.pdf

what were they used for? As the name suggests they were used for fishing. Either trapping or redirecting fish to make them easier to catch.

I am curious do you mean a dam that powers electricity? Or do you just mean a dam, or structure that diverts water?

We've probably been building dams for as long as we've been humans.

We know that Aboriginal Australians would dam watercourses and their civilisation may be as old as 50,000 years.

We think they did it to manage fresh water supplies.

Source: "An Aboriginal Dam in Northwest New South Wales", Rowlands and Rowlands, Mankind vol.7 (1969) pp132-136