The Executives empowered to carry out Roman laws were called Consuls.
When Rome became a Republic it elected two Consuls to form its head of government. These Consuls had many functions, among them the power to carry out laws. Consuls were elected every year. The Senate nominated the candidates and the citizens voted.
The Consuls held absolute authority over all matters of state and could only be questioned by each other. They oversaw military affairs, criminal law, and even the Senate.