Answer:
   
   Most Conservative rabbis hold that playing a musical instrument on
   Shabbat is a violation of a rabbinic law, and thus prohibit it. Some
   Conservative rabbis hold that it may well not be a violation of
   halakha (Jewish law), and so theoretically could be permitted.
   However, in practice, the majority of these rabbis do not permit this
   use. A minority of Conservative synagogues do allow the use of an
   organ on Shabbat, and they have an official teshuvah (responsa) from
   the Rabbinical Assembly's Committee on Jewish Law and Standards (CJLS)
   to back them up. This teshuvah does not reccomend the use of an organ
   - in fact, it discourages it. But it notes that strictly speaking,
   this practice does seem to have some legitimacy.
   
   The website of the Conservative synagogue (below) explains why it
   allows the use of an organ during Shabbat services. It summarizes the
   halakhic arguments in this issue.
   [5]http://www.bethyeshurun.org/organ.htm