Do this words jail vs Jail means exactly the same thing?
I’m sorry, the words are: prison vs jail 
Yes they do.
If they mean the same thing, when we use one, and when we use another? Both are formal?
The word jail is not commonly used in the UK and when we do we spell it gaol.
I would say jail is slightly less formal, though in everyday conversation it doesn’t really matter which one you choose to use.
In the US a jail is somewhere you’re sent for less than a year for a crime and prison is more than a year…
To me, jail and prison are completely interchangeable although as Nathan says, prison is slightly more formal.
The wikipedia says that in most of the countries jail and prison are synonymous words. So, in united the states it’s different. It’s like what blindside said.
The most notable difference is that prison inmates have been tried and convicted of crimes, while those in jail may be awaiting trial. A prison is under the jurisdiction of either federal or state, while the jail holds people accused under federal, state, county and/or city laws. A jail holds inmates from two days up to one year.
Just use “penitentiary” and be done with it. Although, it’s generally best to avoid any of these places.
We don’t understand the word penitentiary in the UK.
We do understand “being banged up” to mean being imprisoned. That may mean something different in the States though 
Ahem, it does mean something different, actually. To bang something up means to damage, as in “I really banged up my foot.”
This reminds me of the good old “knock me up” joke.
Will you get put in the slammer if your jokes are too near the knuckle?
Whoa, had to look “near the knuckle” up. Back to the “knock me up” joke, though. I first read it in The Mother Tongue by Bill Bryson:
“Some of these can cause considerable embarrassment, most famously with the British expression ‘I’ll knock you up in the morning,’ which means ‘I’ll knock on your door in the morning.’”
But, probably, my favorite is by Mr. Arthur Conan Doyle:
“Very sorry to knock you up, Watson,” said he, “but it’s the common lot this morning. Mrs. Hudson has been knocked up, she retorted upon me, and I on you.”
That’s just . . . gross, you guys. Sherlock should have been put in the click for doing this to Watson and Mrs. Hudson.