I'm not a lawyer, but here are the types of scrutiny that I am aware of:
Strict scrutiny is the strictest scrutiny that can be applied, and that's good for us. According to Wikipedia and my slight previous knowledge, if strict scrutiny is applied in this case then for the machine gun ban to be upheld:
It must be justified by compelling government interest. Basically, the government can't just ban machine guns because it wants to. The government has to have a very good reason for doing so.
The law must be narrowly tailored. I'm a little less sure of the details of this criteria, but basically the law can't be overly broad and must be specifically designed to meet the government's compelling government interest goal and no more. It can't address the government's compelling interest and then also do a bunch of other stuff.
The law must be the least restrictive means for the government to achieve its goal. This is tied to the compelling government interest criteria. Basically, let's assume that the reason the government wants to ban new machine guns is to reduce crime. Considering the very few crimes that have ever occurred with machine guns, this reason might not even pass the compelling government interest test, but for the sake of argument we will say that it does. In that case, for the law to be least restrictive, banning machine guns would have to basically be the least the government could do in order to prevent crime. Even if the court finds that banning machine guns prevents crime and that preventing crime is a compelling government interest, the court might still decide that banning machine guns is going overboard and that the government could do something a little less drastic to achieve the goal of preventing crime. In that case, the law would get thrown out because it was not the least restrictive means of achieving the government's goal.
Strict scrutiny is very strict. Intermediate scrutiny is a little less strict. Basically, the government needs an important reason for the law, but not a compelling reason. The law won't stand simply because the government wants it to, but it will if there is an important reason for the law's existence and if the burdens of the law and the rights of the defendant aren't too out of balance.
Rational basis is the least strict means of scrutiny. Basically, the court asks if there is a rational reason for the law's existence. That's a pretty low bar because everything has some sort of reason.