The word jubilee entered English from Old French, but French got it from the late Latin jubilaeus. The Latin word is linked to the even earlier Hebrew yōbēl, which originally meant 'ram', then 'ram's horn used as a trumpet', which was blown to proclaim the jubilee year (so says the OED).
The first sense of 'jubilee' in the Oxford English Dictionary relates to Jewish history. The full term is 'year of jubilee', and refers to a 'year of emancipation and restoration' held every fifty years and announced by a blast of trumpets. During the year of jubilee, according to the OED, "the fields were to be left uncultivated, Hebrew slaves were to be set free, and lands and houses in the open country or unwalled towns that had been sold were to revert to their former owners or their heirs".
There is also a specifically Roman Catholic sense of jubilee: "A year instituted by Boniface VIII in 1300 as a year of remission from the penal consequences of sin" (OED).
In its meaning of 'anniversary', jubilee on its own first meant a 50-year anniversary. In the late 19th century people started talking of a silver jubilee (25 years), after the pattern of 'silver wedding', which had been coined a few decades earlier. Diamond jubilee was coined on the same pattern.
Diamond jubilee is mentioned at the 'jubilee' entry, but the OED has not yet caught up with 2012 events and still has the definition "applied to the celebration of the sixtieth year of the reign of Queen Victoria." The OED is so huge that it takes years, sometimes decades for entries to be updated.