In general, the role of the legislative branch is to create laws and the role of the executive branch is to carry out the laws the legislature makes. This is the case with the No Child Left Behind law as well.
Broadly speaking, the role of Congress was to create the basic NCLB law. For example, it was Congress that laid out the idea that adequate yearly progress must be made by each school every year. It also was the body to set the goal of having all students at grade level by 2014.
Once such a law is made, it is up to the executive branch to administer it. In this case, it has been the Department of Education which has been in charge. For example, it is the Department of Education that has to decide when to allow states to have waivers that release them from the NCLB's requirements.