It depends a bit- there are many things in the world apart from science. Falsifiability is important- I like to describe it as testable rather than falsifiable, though. Anyway, so if something is not falsifiable, what is it? Well, there are a few options:
1- Philosophy: philosophy is probably the most "legitimate" non-scientific field. I find it a bit wishy-washy myself, but philosophy does aim to promote rational discussion and analysis of issues that can't be tested. E.g. philosophy might inquire about what is the right thing to do, which can't be tested.
2- Opinion: some things are fairly subjective and can't be tested scientifically. For instance, someone might have an opinion on say, the healthcare system. In the USA, you need to pay to see a doctor and can't see one if you can't afford to, which some people consider unfair (in Australia, for instance, everyone gets access to a doctor for free if they need it). There's no real way to test what is and isn't fair.
3- Religion: religious belief systems are usually not falsifiable because they attribute any scientific concerns to "God did it" or "the devil is deceiving you" or simply "we don't care what the science says". While many of their propositions can be falsified scientifically, most religions don't want to know. And it's not really impossible to "disprove" God- if he's omnipotent with unlimited powers, any test you try, he could presumably just magic it away. It's not exactly convincing but that's how they try and sell it.
4- Pseudoscience: pseudoscience is when something is promoted as science and often uses scientific-sounding terminology but isn't actually legitimate science. Some pseudoscience is falsifiable but simply rejects any attempts to do so. Polygraphs (lie detectors) are a good example- there is no scientific evidence that they work, but their operators keep using them.
As for "Last Thursdayism", it's basically a joke belief system designed to show how falsifiability is important. The premise of Last Thursdayism is that the entire universe was magicked into existence last Thursday, with everything exactly as it is- all the fossils already in place, all the carbon isotopes set up exactly to look like they are thousands of years old, all the memories burned into people's minds, everything was magically created exactly as it is.
This belief system is not falsifiable because there is no way to test it. If everyone was created with all their memories as they are, thinking they were many years old, then there is no way to tell the difference between this and actually being many years old. Any test you come up with, you could just say "nope, the universe was created last Thursday with those results already there". So it's not falsifiable because there's no way to test it. It's clearly an absurd claim, but it can't be falsified.
Last Thursdayism was created as a sort of counter-argument to the claims (usually religious) that something such as God "cannot be disproved" as though that is enough. Religious people sometimes say that because you can't disprove God's existence, that's enough for them to believe he does exist. Joke belief systems such as Last Thursdayism were created as a way to highlight the absurdity of this argument, and show that just because something can't be disproved, it shouldn't be accepted as true.