Just went through this. There are two basic types of technologies - GSM and CDMA. You want a GSM phone which is used in Peru (and most of the rest of the world) and can take chips. US carriers such as AT&T and T-Mobile operate on GSM and phones used on their networks should be able to easily be unlocked and used in Peru. Verizon and Sprint are CDMA and their phones will not work in Peru. iPhones (4s and more recent) can operate on both GSM and CDMA.
I got a Verizon Samsung Galaxy S3 as a gift while in the US, but I opened it up and found that the chip is smaller than the normal SIM card normally used in Peruvian cell phones. Is there any chance that Movistar has a compatible chip for this phone?
With Claro at least you get normally a 'convertable' chip, which basically is a nanochip, with on the plastic where the chip is located has two shapes which allows you to change it to a micro-sim or a standard sim. So they are for any cell phone. I suppose Movistar has something similar.
If you get a sim card that does not fit the slot in your phone (micro or nano) than you can manually cut the sim card to size. There are various (you tube) tutorials on how to do this. I have done it several times for fooks with upgraded their phone but kept there old sim card. It literally takes 2 minutes to cut.