"The exact definition of Jewishness is not universally agreed upon—neither by religious scholars (especially across different denominations); nor in the context of politics (as applied to those who wish to make Aliyah); nor even in the conventional, everyday sense where "Jewishness" may be loosely understood by the casual observer as encompassing both religious and secular Jews, or religious Jews alone. This makes it especially difficult to define who is an Ashkenazi Jew. The people have been defined differently from religious, cultural, or ethnic perspectives.
Since the overwhelming majority of Ashkenazi Jews no longer live in Eastern Europe, the isolation that once favored a distinct religious tradition and culture has vanished. The center of Judaism today is once again in Israel, although a large community continues to live abroad, particularly in the United States of America, where Ashkenazi Jews live alongside other Jewish groups. Furthermore, the word Ashkenazi is often used in non-traditional ways, especially in Israel. By conservative and orthodox philosophies, a person can be considered a Jew only if his or her mother was Jewish (meaning, more specifically, either matrilineal descent from a female believed to be present at Mt. Sinai when the ten commandments were given, or else descent from a female who was converted to Judaism before the birth of her children), or if he or she has personally converted to Judaism. This means that a person can be Ashkenazi but not considered a Jew by some of those within the Jewish communities, making the term "Ashkenazi" more applicable as a broad ethnicity which evolved from the practice of Judaism in Europe."
-Wikipedia, Introduction to Ashkenazi Jews