What is the X in X-mas?
In a central processing unit, a register in which intermediate results are stored. Without an accumulator, it would be necessary to write the result of each calculation (addition, multiplication, shift, etc.) to main memory and read them back. Access to main memory is slower than access to the accumulator which usually has direct paths to and from the arithmetic and logic unit (ALU).
The canonical example is summing a list of numbers. The accumulator is set to zero initially, each number in turn is added to the value in the accumulator and only when all numbers have been added is the result written to main memory.
Modern CPUs usually have many registers, all or many of which can be used as accumulators. For this reason, the term "accumulator" is somewhat archaic. Use of it as a synonym for "register" is a fairly reliable indication that the user has been around for quite a while and/or that the architecture under discussion is quite old. The term in full is almost never used of microprocessor registers, for example, though symbolic names for arithmetic registers beginning in "A" derive from historical use of the term "accumulator" (and not, actually, from "arithmetic"). Confusingly, though, an "A" register name prefix may also stand for "address", as for example on the Motorola 680x0 family.