To "insist" means something halfway between "asking" and "ordering". When you "insist" on something, it means you won't accept "no" for an answer.
For example, when talking about the bill at a restaurant, you can say this to a friend:
A: Let's split it.
B: No, it's on me. I insist.
This means that you absolutely want to pay for it.
You can also describe exactly what a person has insisted:
She insisted that I take the money, so I took it.
Notice that the verb is in the present tense ("take") even though the sentence is in the past tense ("insisted"). Here's another example:
We told them that we would get a hotel, but they insisted that we stay with them.
Another way to use "insist" is "insist on ___ing":
Why do you always insist on making everything so complicated?