Of course, Westeros is the main stage for A Song of Ice and Fire, but how important is it at a ... planetary scope?
There is a lot of civilisation beyond the Narrow Sea and, especially around the Narrow Sea, most of it is very evolved in terms of society and economy. Also the Dothraki may live a very wildling-ish life, but they appear neither afraid of nor foreign to higher cultures. There are also the Wildlings north of the wall, which also seems to be a large civilisation.
So how important is Westeros to the entire planet? Are they a few amongst many or are they the most sophisticated society on that planet and take a rather leading role?
More information I found
I rewatched the TV show and did a little army counting on the side. The numbers don't feel mathematically sound, but maybe the Westerosis are very good at accounting for losses in most recent battles, while I'm not. Give or take a few garrisons, all of Westeros has (had) about 300k soldiers. If I had to guess I'd say that's a little more than the Wildlings and all of Essos put together, but I can't find solid numbers on those. So, measuring in military size and assuming there aren't any other big inhabitated continents we don't know of, Westeros is about half the world. I'm still having trouble gauging the size of Essos.
In Yunkai alone, there are about 200k slaves (according to TV show S3E7). This allows for speculation that Yunkai alone has a population somewhere between 200k and 1m people. If Yunkai does not greatly outsize all other Free Cities, Essos probably has a much larger population than Westeros. Kings Landing has a population of 500k.
Even more information I found
Just reading A Strom of Swords and it would appear Mereen, Yunkai and Astapor are not of the nine Free Cities. IIRC, I got this impression from the Daenerys chaper where she talks to Kraznys mo Nakloz for the first time. Comparing impressions from reading the books vs. the TV-series, according to the book there are way more Dothraki and other tribes/people living on that side of the narrow sea. So in terms of population I would reduce my guess, making Westeros about one third of the known world.
In terms of society and technology I also wouldn't call Westeros leading: Westeros: Their believes and knowledge seem to have devolved over time. There are the new gods and people somehow accept there are old gods as well. Everything else is little more than folly, if at all. The Maesters can choose to study magic but few do and none can actually perform magic. The alchemists seem a pathetic guild of engineers. Nobody believes there are things like mamoths, giants, skinchangers and such. The remaining acknoledged thing are the Children of the Forest, but even here everybody knows there aren't any left anywhere. Essos: Many believes practiced more and less, but people are hardly judged by that. Maegi are more than tales and everything that may seem impossible/unrealistic can be done by at least one person from Asshai. If not, you will still find at least one person who will claim they once met someone from Asshai who ... The Warlocks in Qarth live more ore less openly among other people. In general. the Free Cities appear bigger not only in population and size but also the architecture, than any dwelling in Westeros.
All of Essos seems well arranged, even settled, while all of Westeros goes to war whenever a highborn firstborn hits puberty.