A register machine is a hardware or software unit that when working with data takes it from memory, puts it in a location where it can work with it quickly, and then returns the result.
For example a regular CPU is a register machine. Since the ALU (the unit that works with numbers in a CPU) can only work with numbers in a register.
A stack based machine adds the data onto a stack and then either pops or pushes stuff onto it.
For example, adding two numbers would be
Push 2 // Push 2 onto the stack
Push 3 // Push 3 onto the stack
Add // Add the top two things on the stack.
When in a register machine it would be something like this.
Load x, r0 // Load x onto register 0
Load y, r1 // Load y onto register 1
Add r0, r1, r2 // Add 1 and 2 and store the result in register 2