Who is a Jew? Who is Jewish? How Jewish are you?

There are communal identifying factors, Halakhnocratic factors, and personal identification factors involved here. Within the Jewish world I have found that the most important are those factors by which you are openly identifiable as a Jew. Lacking any one of these is a strike against you in the orthodox community as well as many other factions. Without further ado, they are:

Jewish by Ethnicity
Jewish by Culture/heritage
Jewish by Religion

Communally, ethnicity is simple: Jewish ethnic/genetic heritage undisputed by communal memory from time immemorial. For the Halakhnocrat, proven matrilineal descent is required. Personal identification as a member of the tribe may require no more than the fact that there is one Jewish parent and that ones family history is that those of the Jews. Often though the genetic factor is more about perception than anything else of course; Jewishness of looks, Jewishness of name, family relatives known to be Jewish, etc. 

Culture is a bit more interesting. Culture is hard to define, but it may express itself in ritual, psychology, patterns of speech, mannerisms, and type of profession. One not dressed conspicuously Jewish and living an assimilated lifestyle, acts like woody Allen (in his films), has a Jewish name, and doesn’t engage is too many gentilesque pastimes may be considered as such. But there are many other more realistic variations. There are people who may not be ethnically Jewish to Halakhnocratic standards but have one Jewish parent who comes from an assimilated family and nonetheless values their heritage to some degree may be someone who considers himself or herself Jewish by heritage but not ethnically or religiously so. 

Jewish by religion is not completely obvious. Someone who can be identified as Jewish due to his or her conspicuous observance to religious Judaism not just in terms of occasional ritual performance, but also by the persistent projection of their religiousness through their form of dress is identifiably Jewish by religion. Halakhnocratic standards obviously must apply to complete the picture. A person like this though may be a Baal Teshuvah new to orthodoxy still conspicuously culturally non Jewish—the same may apply to the convert. 

More specifically though: One can be religiously Jewish but neither culturally nor ethnically. Though it is possible to be religiously and culturally Jewish without the ethnic factor ethnically. It goes without saying that one can be ethnically Jewish without being culturally or religiously Jewish, just as one can be culturally Jewish but not technically ethnically or religiously—but that is more an issue of Halakhik nuance. 

I am not ethnically Jewish, but I am both culturally and religiously Jewish, having been raised from birth essentially within the Jewish community and within orthodox schools and by continuing to be conspicuously orthodox. 

This model has been important to understanding my identity as a Jew.