Ashkenazi Jews are Jews who:
a) settled in Germany
b) came to Eastern Europe and Eurasia from Byzantine
c) fled from Spain and Portugal during the Inquisition.
In my case, for example, my dad's autosomal DNA (with AncestryDNA Version 2.0) indicates that our case is b) and c). My cousin Kevin, however, got matches in Germany for the R1a1a1 YDNA marker, but this is probably because we have Foczko relatives who moved there. The Foczkos are Ashkenazi Levites; and, like the rest of the 52% of all Ashkenazi Levites, the Foczko men have the R1a1a1 marker. I do not know where the Rusznaks, who were also Ashkenazi Levites, fall; as none of the Rusznak men, as far as I know, have ever taken a YDNA test to see whether they have a R1a1a1 (Khazar or Slavic), J (Jewish), or E3b (Jewish) marker.
The atDNA of Dad, nonetheless, heavily indicates Eastern European/Khazar admixture, as I should have originally expected. Also, I know that it's Khazar instead of Slavic because I have the ADH2*2 allele—I know this because I get sleepy after one glass of wine (which is nice, given that my paternal granddad's paternal granddad—whose tribal origin we do not know—was an alcoholic. By the way, his mother's surname doesn't help, either; as "Daniłowicz" simply means "ben-Dani'el". I had hoped that "Daniłowicz" might indicate tribal affiliation with Dan, but it doesn't.).
For more information, see Khazaria.com.