absolute error

Absolute error is the plain, no-frills distance between an approximation and the true value: if x is the true value and x~ is your computed approximation, the absolute error is |x~ - x|. It answers the simple question 'how far off am I, in the same units as the quantity itself?' A length predicted as 5.03 metres when the truth is 5.00 metres has an absolute error of 0.03 metres.

Because it carries the units and the scale of the quantity, absolute error is exactly what you want when there is a natural tolerance in those units — 'the part must be within 0.1 mm', 'stop iterating when the change is below 1e-8'. For a vector or function you measure it with a norm, for example ||x~ - x||_2 (the Euclidean length of the error vector) or a max norm (the largest single component error). In an iterative method, the change between successive iterates, |x_{n+1} - x_n|, is a cheap practical stand-in for the true absolute error you cannot see.

Absolute error has one blind spot: it does not know whether the quantity is big or small. An error of 0.03 is tiny next to a true value of a million but catastrophic next to a true value of 0.001. That is precisely why relative error exists, scaling the error by the size of the answer. Use absolute error when the natural scale is fixed and known; reach for relative error when the magnitudes span many orders, as floating-point numbers do.