Executive orders are the President directing the executive branch how to carry out their jobs.
Executive orders can be illegal if they direct the executive branch to either do something illegal or if they direct the executive branch to fail to carry out their mandated duties. In that case the executive order may be challenged in court -- that is one of the things that is going to happen with this executive order. There will be cases flowing up the court system of people claiming this order is illegal, discriminatory, and violates the rights of legal residents.
Executive orders can also not be illegal, but may be taking advantage of flexibility that Congress regrets giving the executive branch. For instance, if the President begins selling off national parks, that may actually be within his authority, but it might inspire Congress to pass a law prohibiting him from doing so in the future.
As to how Trump is doing so many radical things so quickly, I would simply point out that he likely is not worrying about the consequences.
The President can issue executive orders very quickly if he chooses, but when most Presidents issue one creating a new, radical change (as opposed to the several dozen executive orders that get signed every time control of the White House passes between the two parties, like the Mexico City Rule), he first will spend a lot of time working with the affected departments and with his own lawyers to figure out exactly what he has the authority to do. For instance, before Obama introduced DACA, he first had to spend a lot of time figuring out exactly how much of the DREAM act he could get away with implementing purely with his executive power -- he didn't want to take the action and immediately have it reversed by the courts.
Now, if you don't worry about what you can and can't do and just do it, everything moves a lot faster. You might say Trump is governing the Facebook way: Move Fast and Break Things.