According to Jewish law, the child of a Jewish woman is considered Jewish, regardless of whether that child's father is Jewish or not. Any child of a non-Jewish mother is considered non-Jewish, unless he or she converted in accordance with Jewish law (the halacha).

When Israel was declared as a state in the 1940's the new state's laws were a mixture of new laws, English law (due to the fact that the UN had given England a mandate to administer the area of what is currently Israel, as well as the West Bank and the Gaza Strip), and Jewish law. The last was, and still is to this day, the law of the land as far as marriage and burial for Jews. For non-Jews those laws obtain from their own religious background.

There are various cultural problems that result from that provenance of Israel's laws. Since a majority of Jews in Israel are secular (i.e. not religiously observant), forcing them to comply with Jewish religious law in order to get married, or to bury their deceased family members can and often does raise major issues. For example, according to Jewish law, a cohen (one whose lineage is from the priests of ancient Jewish times, typical family names include Cohn, Kahna, Katz, etc.) may not marry a divorcee. Thus, if a secular Israeli Jew of priestly descent falls in love with a divorced woman (even if she is Jewish), the two cannot marry in Israel. Indeed, in many such cases the couple travels outside the country to get married, and after their return to Israel, the law does accept them as married, through the general reciprocity with other countries' laws. There are many other examples, but this one suffices to illustrate the issue of current Israeli law vis a vis Jewish law.

According to the Israeli "Law of Return" ("chok ha'shvut") any Jewish person who wishes to immigrate to Israel may do so. For the purpose of this law, the legislature decided to err on the side of inclusion and considers anyone who had at least one grandparent (of either gender) who was Jewish as Jewish *for the purpose of immigration*. However, once in Israel, if this new immigrant's mother was non-Jewish, this person is not considered Jewish for any other purpose. If this new immigrant or any of his/her children then fall in love with a Jewish person, well, you get the idea...

There are indications that this matri-lineal passage of "Jewishness" traces back to biblical times (see e.g. the excellent source quoted by @bunnyphuphu - http://www.beingjewish.com/identity/whoisajew.html ), there are also arguments that go back only a few centuries to the time of "pogroms" or attacks by gentiles on Jewish towns in Eastern Europe and Russia. In these attacks, many Jewish girls were raped by the attackers, and lacking modern contraception, often became pregnant with the babies of their rapists. By determining that the child of a Jewish woman is Jewish, regardless of the father's religion, the children of these tragic cases could be considered part of the Jewish community, and be raised by their mothers as Jewish.

The bottom line is that while some Jewish communities (non-orthodox) may have decided not to follow the halacha in this matter, Jewish law specifically states that you can only be considered Jewish if one of two hold true for you:

1. Your mother was Jewish.
2. You completed a conversion to Judaism according to the requirements of the halacha.

This is the law of the land to this day in Israel (except that for immigration purposes it is not applied).

This is still a touchy subject, and you'll get a different answer depending on how Orthodox they are.
This link I found really sums up the issue between the Orthodox and Non-Orthodox views.
http://www.beingjewish.com/identity/whoisajew.html

My best friend growing up had a father who was Jewish and a mother who was not.
She considered herself half Jewish.

wy, you specifically asked about adherence in "Israel or Jewish communities". There are differences between the two - not to mention sub-differences and sub-sub-differences and, well, you get the idea. I'll try to get some up-to-date info and come back later.

1. 1st law: Child's non-Jewish mother:: A child born of a non-Jewish mother should be considered Jewish through the father's Jewish identity.
2. 2nd law: Child's non-Jewish mother:: A child born of a non-Jewish mother should be considered maternal identity.
3. Mixed marriage in Jewism is strictly forbidden as is the union of a horse and a donkey, and in both unions the offspring are judged matrilineally.

……………………..quote…………………
According to the Mishnah, the first written source for halakha, the status of the offspring of mixed marriages was determined matrilineally.
According to historian Shaye J. D. Cohen, in the Bible, the status of the offspring of mixed marriages was determined patrilineally. He brings two likely explanations for the change in Mishnaic times: first, the Mishnah may have been applying the same logic to mixed marriages as it had applied to other mixtures (kilayim). Thus, a mixed marriage is forbidden as is the union of a horse and a donkey, and in both unions the offspring are judged matrilineally. Second, the Tanaim may have been influenced by Roman law, which dictated that when a parent could not contract a legal marriage, offspring would follow the mother.
http://en.wikipedia.org/wiki/Who_is_a_Jew%3F
………………/quote…………………….

en.wikipedia.org/wiki/Who_is_a_Jew%3F -

Literally, it's called Judaism 101:

"Who is a Jew?

"A Jew is any person whose mother was a Jew or any person who has gone through the formal process of conversion to Judaism.

"It is important to note that being a Jew has nothing to do with what you believe or what you do. A person born to non-Jewish parents who has not undergone the formal process of conversion but who believes everything that Orthodox Jews believe and observes every law and custom of Judaism is still a non-Jew, even in the eyes of the most liberal movements of Judaism, and a person born to a Jewish mother who is an atheist and never practices the Jewish religion is still a Jew, even in the eyes of the ultra-Orthodox. In this sense, Judaism is more like a nationality than like other religions, and being Jewish is like a citizenship."

From what I've seen, it's really more of upbringing, at least in the US.