是定语从句
在"it contained half the money (that) he had lost”中,"(that) he had lost"是定语从句,用来修饰"money"(或限定"money"的范围),其中that可以省略.
补充:
关系代词that在从句中做宾语,就可以省略.如:
the flowers (that) I bought yesterday.
that指代flowers,在从句中做bought的宾语,就可以省略.
做主语的话就不能省.如:
All that glitters is not gold.发光的不都是金子.
that在从句中做glitter的主语,所以不能省.