When we talk about "the interest rate" that government bonds move inversely to which is this rate?
Bonds move inversely to their own yields, which are calculated from the bond prices themselves. It sounds circular, but it just means prices and yields are two alternative ways of looking at bonds. A government bond's yield is often considered the market's expectation for the average of short-term rates (as set by the central bank, noted by Mike Scott) over the term of the bond. Thus, importantly, changes in the short-term rate that are expected (due to known economic conditions or previously telegraphed hikes or cuts) do not directly affect the price or yield of a given bond. The yields plotted from the shortest- to longest-term bonds form the yield curve, which indicates the market's interest-rate expectations.
In general, it’s a benchmark short-term interest rate set by that country’s central bank (or to be more precise, the central bank for the currency in which the bonds are denominated). The name and exact details will be different for different countries. In the UK, for example, it’s the official bank rate set by the Bank of England.