Generally The East/West divide was centered on the lineages of nations from the side of Imperial Rome they fell under when it went the two empires system to better manage the large territory the empire held occuring sometime around 395 AD. This split largely was divided on the common languages spoken by the citizens. West of the Adriatic was Latin dominated, while East of the Adriatic was largely Greek. Both Empires were called the "Roman Empire." The Western Roman Empire fell shortly after this split, the Eastern Roman Empire endured for sometime (for the sake of my sanity, I'll be referring to this half of the Roman Empire as the Byzantine Empire, even though until the very dying days of Byzantium, they still thought themselves as Romans.). It's from here the earliest East/West divide of Europe came into being and was largely seen as "The West" to mean west of the Adriatic sea and "The East" (Orient) is east of the Adriatic. Byzantium would endure for almost a millennium longer than the Roman Empire thanks to the strategic location of Constantinople. Which is now Istanbul, not Constantinople. When it was suddenly gone, the name Constantinople. Why did Constantinople get the works? In 1453 it was captured by the Turks.
Sorry, couldn't resist.
At either rate, when Istanbul became nobody's business but the Turks, it cut of Western Europe from trade to the east (Eastern Europe, the "middle east" and the "Far East" via loss the world's only Trans-continental city, which sat on the Bosphorus straits, the connection between the Mediterranean and Black Seas.). Because of this new problem, many Western Europeans began looking for ways to continue trade with Asia but not have to deal with the Turks. This gave way to a man for Geona getting financed to Sail West to Go East, and thus Christopher Columbus became the most famous man in history who's notable achievement was not going to where he wanted to go... which lead to new riches for the west as "The New World" did not have people living there... well... no Catholic people... but when you're a Western Europe power in the 16th century, if you're not some kind of Christian, are you really a person?!
With that said, who is "West" and who is not "West" has generally fallen into this catagory. If you have cultural ties to Western Roman Europe (This includes the Nordic, Germanic, and Anglo nations) or are/were a colony there of (all of the Americas... including LATIN America, Israel, Australia, and New Zealand) than you're generally "West". If you don't, than your the rest.
That said, it's not entirely clear as it also tends to be drawn on political lines. Generally, the West today includes Eurasian and American powers that are either in NATO or really want to be part of NATO (not Turkey, which is only there because it allowed the U.S. to have a place to stage attacks against the Soviet Union. As a whole, Turkey desperately wants to be considered European... east or west... but Europe generally doesn't want them to be considered European) or EU. This does encompass a lot of Non-Europe and Non-Americas into it (such as Singapore, Oceania, various British or formerly British holdings (the Empire that the sun never sat on), though generally not India/Pakistan or South Asian powers. South Korea and Japan are "Western" unless Statistically Convenient (i.e. European nations looking to shame the U.S. by pointing out all "Developed Western Nations but the U.S. have abolished capital punishment" and over look that Japan, Korea, and Singapore are also considered developed nations and retain the death penalty. But they're not Western... despite Singapore being a member of the Commonwealth of Nations! Geography won't save you here either, as Austrailia and New Zealand (which both abolished capital punishment) are considered "Western".). Ukraine of late tends to fall into the Western when coinvent camp.
Other nations vehemently oppose consideration as a Western Power. Despite Russia having far more cultural ties to Europe, they almost always insist they are not "Western" (as does Belarus, which tends to side with Russia a great deal in international politics with exception to the "Belarus is Russian debate" (surprisingly, Belarus supports this position. Russia does not... it's at times adorable.). Historically, Russia's capital has always been West of the Urials a generally considered boundary between Europe and Asia) and generally has been so centralized that at times, Russia east of the capital is considered forgotten by Russians... as is anything west of the Capital... or north... or south... Unless the Capital is St. Petersburg, in which case Moscow is still important. When Moscow is the Capital, St. Petersburg will be important only in that it used to be the capital... or it's current name might be problematic.
In religous matters, "The West" involves Juedo-Christian nations that were Roman Catholic following the Great Schism between but prior to the Protestant Reformation. This generally falls on the same "Two Romes" East/West divide as most of those nations are East Orthodox Catholicism (the big difference between the sects is that 1.) Eastern Orthodox masses are said in Greek while Roman Catholic are said in Latin/Vernacular, save for a tiny portion which is said in Greek. 2.) Roman Catholics have a supreme pontiff (the pope) for leadership while Eastern Orthodox's leadership is a patriarch for each nation, with the patriarch of Constantinople being the "First among equals" leader for the faith as a whole... and they historically had less political power than the Pope had in their own faith. 3.) Statues of Jesus and the Saints: Roman Catholics have no problem with this while Eastern Orthodox sees statues as false idols and it's taboo to make them of holy people. Interestingly enough, modern Roman Catholics and Eastern Orthodox Catholics have less problems with each other than Roman Catholics do with Protestants, to the point that Roman Catholics may receive communion from Easter Orthodox Ministers (however, attending Eastern Orthodox liturgies do not meet mass requirements, so it's generally advised to abstain from Eucharist during Orthodox Churches if one can attend a Latin Right Sunday mass... however in an emergency, the communion is still valid).
In Linguistics, the East/West split is generally based on alphabets used for languages. The Latin alphabet is used by "Western" European Language families even if the language is not a Latin language. Greek or Cyrillic alphabets are used by Eastern European Languages.
Because of the colonial period being participated in by Western nations, the use of the Latin alphabet by non-European languages for it's written form (such as Hawaiian, Quechan, Navajo, to name a few) does not make it a western language, as they are not from remotely close families and only use the Latin alphabet because that was the alphabet of the speakers who turned the formally verbal only languages into written languages.
Edit: Almost forgot, though clearly not what the question was about, in the United States, "The West" generally refers to a region of U.S. territorial holdings in the Continental U.S. that were obtained through expansion and generally considered wild when compared to the Eastern Seaboard. Generally the barrier of "The West" is the Mississippi River in the Modern U.S. and is likely to remain there, with states on the west of the line considered "Western States", though as mentioned this was historically fluid and prior to the final territorial acquisition in Mainland U.S. following the Mexican-American war, the east/west divide tended to be the Appalachian Mountain range which runs through most states that boarder the Atlantic (Florida, being a sandbar with delusions of grandeur, is the only Atlantic coast state that this feature is not in) with the West being territory gained in the Louisiana purchase. Prior to that, the Western border was much closer to shore line. In fact, the reason that the attempt by the British to disarm the citizens of Concord, Massachusetts was the inciting incident of the Revolutionary War was because at the time, Concord was the last bastion of civilization before the lawless west... essentially the British Army was going to disarm the citizens on the threshold of the wild west and return to Boston without any means for them to protect themselves. Lexington is about 19.5 miles from Boston Harbor... with modern transport, it's less than a 45 minute drive and in 1775, the British Army were able to march there and back, while getting hit by guerilla fighters the entire way home, and managed to be back in Boston later that day... just in time time for some 7,000-16,000 strong militia men to lay siege to the city and blocking all egress by land for British forces for 11 monthes.