Is being gay a sin? Can one be a Jew and a gay person at the same time? How does the Jewish community think about gays?
This is a very controversial subject. However, a distinction must be made in terminology between "being gay" (which for the purposes here we could define as being sexually attracted to members of one's own gender) and engaging in homosexual acts (where Male-Male intercourse is an unambiguous violation of a Torah-level prohibition and Female-Female may be a rabbinic prohibition).
So with those definitions, being gay would not be a sin from a Judaism perspective. Engaging in homosexual acts is a sin. One can definitely be a Jew and be Gay at the same time, and one can be Jewish and engage in homosexual acts at the same time, just as one can be Jewish and violate the Sabbath or eat non-Kosher food at the same time - a Jew who sins is still a Jew.
Regarding how the Jewish community thinks about gays - it depends on the community. This can range from being very respectful and welcoming (and for some of the most liberal non-Orthodox denominations, allowing and sanctioning such unions) to don't-ask-don't-tell to downright hostility. It just depends on the community (though the topic is definitely more polarizing than "how does Judaism think about Sabbath violators", or almost any other topic).
@Yaakov Ellis gave an excellent answer, but I would add just a single thought to explain to don't-ask-don't-tell attitude of many communities.
As mentioned before, Judaism sees homosexual acts as a sin. (Perhaps different levels of prohibition depending on the gender.) However, Judaism does not generally consider predilection to a sin to be a sin itself. It may a difficult test, to be sure, but Judaism expects people (of all backgrounds) to overcome their temptations (whatever they may be).
Judaism does not generally approve of broadcasting one's personal tests and shortcomings. (Note that one is not allowed to speak לשון הרע (negative speech) about oneself.) Since Judaism classifies homosexual acts as a sin, one would expect an observant Jews to not broadcast or classify themselves based on their temptation to sin. I think this is the general basis for the don't-ask-don't-tell attitude of many communities. They might be welcoming of gay individuals, but at the same time not be supportive of Gay Pride movements.
The Torah contains prohibitions against many activities that Hashem assures us are spiritually harmful to us. These are mainly activities that we might otherwise want to do.
On one hand, practicing homosexuality is one of these proscribed activities, on the other hand, Jews who are homosexual will find this prohibition far harder than any other.
Certain less tolerant communities will reject a practicing homosexual, as they would reject any Jew who didn't keep Shabbat or kashrut. Communities who are more accepting are more likely to welcome homosexual Jews into their community, even if they don't approve of what that individual does in the privacy of their bedroom.
"Can one be a Jew and a Gay at the same time?"
The way of life prescribed by the Torah is intended for all Jews. Having said that, being a religious Jew I'm sure is exceptionally challenging for a homosexual. But for a Jew who wants to realise their spiritual potential, there's no other way.
Please see Wikipedia's article on homosexuality and Judaism.
Here's my summary of some points from the Orthodox Judaism section of that article.
Orthodox Judaism basically forbids male homosexual conduct.
The Lubavitcher Rebbe and R' Jonathan Sacks both advocate offering help and/or compassion to homosexual Jews.
R' Norman Lamm says that some (but not all) homosexuals are diseased, and need compassion and treatment, not ostracism.
A statement, mainly by Rabbis Nathaniel Helfgot, Aryeh Klapper, and Yitzchak Blau, adds that exclusive homosexual Jews normally shouldn't marry someone of the opposite gender.
The JONAH organization focuses on "prevention, intervention, and healing of the underlying issues causing same-sex attractions." Atzat Nefesh is similar.