Short answer is that Ashkenazi Jews are, roughly speaking, those who can trace their recent ancestry to Northern Europe, and they make up the majority of modern-day Jews.

The term "Ashkenazi" comes from the Biblical name Ashkenaz, used first in Genesis as the name of a person and in Jeremiah as the name of a kingdom. It was not uncommon in the medieval period for the European places where Jews began to settle and form communities to be given names based on the nation names in the Bible, and in this regard, Ashkenaz/Germany was joined by Tzarfat/France and Sefarad/Spain.

That said, while I say Ashkenaz=Germany, more accurate would be to say that Ashkenaz referred to the Rhineland, an area that included parts of both Germany and France. Though the origins of Ashkenazi Jews are still a matter of some controversy (some more and some less valid), the generally accepted view among scholars of Jewish history is that they made their way to the Rhineland from the former western Roman Empire, notably Italy, as traders, and were invited by Charlemagne and his successors in the Carolingian Empire to settle in these lands. The community in the Rhineland at this point (pre-Crusades) was quite small and homogeneous, though scattered throughout a number of towns and villages in the area, and is still known to this day as having been an important center of Torah study as the home of Rashi/Rabbi Shlomo Yitzchaki, probably the most significant Jewish exegete of the Torah and Talmud. At the same time, while the community had begun in the Rhineland, it began to spread into other parts of France and Germany, and in 1066 Jews crossed into England with William the Conqueror.

Not long after this, the Crusades began to cross Europe, destroying many of the original Rhineland Jewish communities in their path; while Ashkenazi communities still remained, their numbers were severely depleted and for the next few hundred years would be constantly in jeopardy, whether due to persecution, expulsion (the Jews of England were expelled in 1290, those of France in 1306), or mass murder- the false blaming of the Jews for the Black Death in the 14th century made that time period an especially hazardous time for Ashkenazi Jews in West and Central Europe. While all of this was going on, many Ashkenazi Jews found that Eastern European lands, like Poland-Lithuania and Hungary, were more hospitable and chose to travel there, causing those communities to blossom and eventually become the predominant Ashkenazi Jewish lands. With them they brought the Judeo-German language which they had been speaking, and over the next several hundred years in Eastern Europe that language continued to evolve into forms of what became a lingua franca of many Ashkenazi Jews, Yiddish.

While migratory patterns since then have meant that Ashkenazi Jews have ended up all over the world, this is their general origin point. At this point, Ashkenazi Jewish means one of several different things:

1. Genetic origin (as you note), in that Ashkenazi Jews, as an endogamous (marrying within its own in-group) group for hundreds of years in specific parts of the world, developed a genome distinctive from the genomes of non-Jewish Europeans from the areas in which they lived and with markers of Middle Eastern ancestry, including similarities to Sefardi Jews (originating in Spain and then moving throughout the Middle East). That's the kind of thing that would show up as being distinctive on the test that you took. The very endogamy of Ashkenazi Jews has led to its being intensely studied by geneticists due to the many recessive genetic disorders for which Ashkenazi Jews often carry genes, and in fact many Ashkenazi Jews today will take genetic tests before having children to see if they are carriers for Ashkenazi genetic diseases, especially if they plan on having children with a fellow Ashkenazi Jew.

2. Religious practice and culture, as Ashkenazi vs Sefardi/Mizrachi (originating in Spain and/or the Middle East and Asia) Jews will have different practices (called minhagim) that greatly differ from one another and very different cultural experiences. Even within those two broader categories, there are much finer gradations which are probably beyond the scope of an overview post like this (for example, when Jews began to move to Poland/Lithuania from Germany, they considered themselves to be creating a new community minhag, which in many respects actually eventually overrode the original Ashkenazi minhag due to the influence of the new Polish Jewish community), but overall, modern Jews will often distinguish themselves as Ashkenazi or Sefardi for the insight that this gives on their past as well as for how it affects their present.