In Ezekiel 8:14 the prophet is taken up in a vision, and shown women who "Weep for Tummuz." The passage does not explain what that means, although these women are engaged in some form of idolatry. My Bible footnotes tells me it is a "fertility god." I understand this passage is in context with Israel's idolatry, but what are they specifically engaged in when they are "Weeping for Tummuz"?
This is a good question. The information available is very complex, so I have gleaned the most relevent information to answer the question. To begin, @lasersauce made the correct observation that
In this regard, Becking and Dijkistra (1996) provide important historical details. That is, there are similarities with the death of the Canaanite God Baal and Tammuz written in the In the Ras Shamra texts. In one of the studies the writer observes that
In summary, the sources indicate several parallels between Tammuz and the Canaanite god Baal.
According to J. R. Dummelow, Tammuz was “a deity worshipped both in Babylonia and in Phoenicia—the same as the Greek Adonis. He appears to have been a god of the spring, and the myth regarding him told of his early death and of the descent of Istar his bride into the underworld in search of him. The death of Tammuz symbolised the destruction of the spring vegetation by the heat of summer, and it was celebrated annually by seven days of women’s mourning in the 4th month (June–July), which was called Tammuz. This superstition had been introduced into Jerusalem.” (A Commentary on the Holy Bible, pp. 497–98.)
Additional source, covering most of the same information, but more recently printed, although not as plainly laid out. 
 https://www.lds.org/manual/old-testament-student-manual-kings-malachi/chapter-26?lang=eng  http://books.google.co.uk/books?id=NopUi53PC0YC&pg=PA70&dq=death+of+baal+Tammuz+ras+shamra&hl=en&sa=X&ei=UZ20U9SPFMuoyASbzYLIBg&ved=0CD8Q6AEwBA#v=onepage&q=Tammuz&f=false