borrow is låna but lend is låna ut (stress on ut).
The only time you can skip ut from låna ut is if you have the person you're lending something right after the verb: Kan du låna mig hundra kronor? 'Can you lend me 100 SEK?' We don't use ut there because the recipient is right after the verb. We still need it in the construction Jag lånade ut 100:- till henne 'I lent 100SEK to her' where the recipient comes later in the sentence.
I find this really interesting, because when I was living in Minnesota (a state with many people of Swedish descent), I would sometimes hear people use the verb "borrow" when they meant "lend", for example "Would you borrow me a dollar?" It sounded ridiculous to me, but seeing that Swedes use the same verb for borrow and lend, it makes sense.
Maybe, but it may also be that it's natural to confuse the two sides of an action that always involves both of them. I think you can hear this in other English dialects too.
Using similar verbs is not unique to Swedish either, in German it's leihen/ausleihen, pretty much like in Swedish (låna/låna ut) for instance.
"Ausleihen" actually is used in both meanings: http://www.duden.de/rechtschreibung/ausleihen. Overall German has a lot of variation for these terms. To lend could be leihen, ausleihen, or verleihen. Either of the first two also means "to borrow", or more commonly you would use the reflexive form (sich leihen). Or borgen/sich borgen etc.
In Brazil, we reverse the logic, we have the verb "lend" and we add something in front to mean "borrow". Some older people even use the same word for both "borrow" and "lend/loan", "emprestar" which means "to lend", as in "vou emprestar um dinheiro com o agiota" (I will borrow some money from the loan shark). [it sounds weird, btw]. Maybe it's just short for "tomar/pegar emprestado" which actually means "to borrow" (literally "to take lent/loaned").
But if you say "o agiota me emprestou um dinheiro" then it means "the loan shark lended me some money".
Tror det kan vara ett ljudfel här. Ljudklippet för hela meningen låter okej, men om jag håller muspekaren över "vi" så låter uttalet om "ve". Inte ens den bredaste bonn-dialekten uttalar "vi" som "ve" i min erfarenhet.
(summary for the English-speakers!) when I hoovered over "vi" it sounds very much like "ve", which is incorrect. ("ve" means "woe" in Swedish)
"rent a car" would be "hyra en bil". A car you rent isn't quite the same as a car you borrow, since a borrowed anything is usually something you get for free as long as you promise to return it while a rented thing is something you have to pay rent for while you keep it :)