Everyone knows better as a comparative form of good / well.
This plan is better than the old one.
She speaks English better than me.
But better is also commonly used as a replacement for "should", especially in spoken English. (used as a helper verb: also called a modal auxiliary verb)
(had) better + base verb
You had better wear a jacket.
We’d better leave now.
You’d better call her.
They’d better hurry or they’ll miss the train.
This structure is stronger than should. It often suggests there will be a problem or negative result if you don’t follow the advice.
You better not be late.
We better go now.
This is very common in spoken English and works as a direct substitute for "should", but it is more informal.
better off
(in a better condition)
You’re better off staying home.
They’re better off without that job.
better than nothing
(having an imperfect something is better than having nothing)
It’s not great, but it’s better than nothing.
It's an old car, but it’s better than nothing. It certainly beats walking.
better and better
Your English is getting better and better.
Sarcastically (after hearing bad news on top of bad news)Oh great! This keeps getting better and better.
better late than never
(being late is better than not being there)
Sorry I’m late—better late than never!
get better — improve / recover
I hope you get better soon.
My English is getting better.
feel better / be better
I feel better today.
Things will get better.
better at + noun/gerund
She’s better at math than me.
I want to get better at speaking English.
better + noun
We need a better plan.
This is a better option.