Based on a section of the Zohar, it is customary for people to welcome in Biblical guests known as ushpizin אושפיזין into their sukkah. The root אושפיז has a number of meanings in Talmudic Hebrew: an אושפיז or אושפיזא means "a lodging place, an inn" (Megila 26a, Yoma 12a) and אושפיזכן is an innkeeper or host (Sotah 37a, Zevachim 18b). Klein writes that the meaning of "guest" only began in Medieval Hebrew - hence the Zohar quote. In Modern Hebrew, the verb אשפז means "to accommodate" or perhaps more commonly "to hospitalize".
What is the origin of the word? Klein writes:
Med. Gk. hospition, hospetion (=inn), from L. hospitium (=inn; hospitality), from hospes, gen. hospitis (=host; guest), which stands for *hosti-potis and originally meant 'lord of strangers'.
Steinsaltz, on the other hand, says it comes from the Persian asfanj or aspanj, also having the meanings of "inn" and "innkeeper".
This site claims that from Persian the word made its way to Latin and Greek, but I haven't seen any other evidence of that. I think if the Latin and Persian roots are connected, it is more likely that they share a common Indo-European root - ghos-ti or ghostis.
This root, particularly in its development in Latin (and from there to English and other languages) has a number of interesting phenomenon associated with it. First of all, as you may have noticed, this is one of the words that went from Latin to Greek, instead of the other way around. According to this site, this was common during the Koine period. (An earlier Greek root that came from the same IE root was xenos, as in xenophobic.)
Also interesting, is the number of seeming opposites that derive from this root. For example, both the English "host" and "guest" have their origins here. Even more striking, there are words with a positive connotation - hospitable, hospice and hotel, but also negative ones - hostile, hostage and host (as in an army). This was due to a tension in relation to guests: on the one hand they were to be treated kindly, but on the other hand they were strangers and to be viewed with suspicion.
While Judaism had (and has) a very positive attitude toward guests (e.g. hachnasat orchim הכנסת אורחים and the minhag of ushpizin), and we are commanded to love the ger גר - stranger), there are also sources where we are warned to be careful of those wishing to convert. But on Sukkot - I'd say it's better to be hospitable than hostile.