The answer to your question may be different depending on which state you live in. Every state government has a legislative branch, but they do not all call them the same thing.
In our American system of government, we have separation of powers. That means that each state government (and the national government) has three branches of government whose powers are more or less distinct from one another. In general, the legislative branch makes the laws, the executive branch carries out the laws made by the legislative branch, and the judicial branch interprets the laws if people disagree about what the laws mean.
Each state government has a legislative branch, but they do not all have the same name. Most states simply call the legislative branch the “State Legislature.” However, some states have “General Assemblies,” “Legislative Assemblies,” or even “General Courts.” Please consult the link below to find out what name your state uses for the legislative branch of its government.