The separation of powers in the US dictates that only Congress has the power to write and enact law. That model has been copied by states as well, such that only state legislature — not governors — can write and enact law. That is written into both federal and state constitutions; it's black letter law, and not something likely to change so long as the US maintains our constitutional form of governance.
That being said, the Executive branch is charged with executing the laws that Legislatures enact — hence the name 'Executive branch' — and execution of a law often entails a lot of decision-making. Realistically speaking, laws have to be general and abstract. Laws can only be written once, but they must be applicable to an open-ended range of cases spanning an open-ended period of time; it would be impossible to foresee and specify every possible use a law could be put to, and most legislatures don't try. As such, the various departments of an Executive administration have a decent amount of leeway for interpreting given laws to create policy, and can even create policy in the absence of any applicable law if there is (in their view) a clear need for governmental action. This is where 'executive orders' fit in: the president or a governor will decide that something needs to be done immediately, and issue an order that it be done without waiting for the Legislature to act. But as we've seen, executive orders can be overturned by courts, or by the next administration, or by the legislature itself if it decides to pass actual law concerning the issue. Executive orders are not law; they are an administrative exercise of power to determine public policy in the absence of clear established law.
Over the last 30 years or so, we have seen a marked decrease in actual legislation. The US Congress has more or less ground to a halt, largely because the GOP has taken on a fortified defensive position: trying to block what it sees as 'Liberal agendas' without really trying to achieve any direct goals of its own. The result has been that the Executive branch has usurped power from congress, making a large range of decisions by executive order that ought to have been worked out and passed into law by congress. Most decisions in this country are now made by the executive branch and fought out in the courts, with Congress as a bystander; note that the current Senate has primarily concerned itself with appointing judges to aid conservatives in winning court decisions, and has largely put aside debating or bolting on bills that have come up from the House. If this particular trend continues, then Congress will obviate itself, and we will be left with the typical 'banana republic' schema in which the Legislature is a vestigial token retained to give the appearance of representation to an authoritarian system.