I'm new to patents and am a bit confused by the whole system. To my understanding a patent is applied at a federal level. Does this mean someone can go to France, see an invention that is patented their, find out how it works, and patent it in another country that doesn't have a patent for it yet? Do countries respect patents from other countries?
No, for two reasons.
First, if you are copying someone else's invention, then you are not an inventor. Only the inventors (or people who have been assigned the rights by the original inventors) are allowed to validly apply for a patent.
Second, an invention must be novel (among other things) in view of what's available across the world to receive a patent. But if you are copying something that already exists, then the invention would not be novel. Thus, if a product is available in one country, it cannot be patented anywhere in the world (excepting grace periods, which are a little outside the scope of the question).
That said, as a minor historical note, in countries which derive their patent law from the UK, there used to be the concept of invention-by-importation. If you were the first to import a product, and the product was novel for that country, you could receive a patent for it. However, this sort of thing hasn't been available for very many decades.