forward error
This is the error you would naively want to measure: how far is the answer my computer gave from the true answer? You aimed an arrow at a target; the forward error is simply the distance between where the arrow landed and the bullseye. It is the most intuitive notion of accuracy — and, slightly counterintuitively, often the hardest to compute directly, because to know the true answer you would have to already have it.
Precisely, if the exact solution is y = f(x) and your algorithm produced y-hat, the absolute forward error is ||y-hat - y|| and the relative forward error is ||y-hat - y|| / ||y||. It measures the wrongness of the OUTPUT. Compare this with backward error, which instead asks how much you would have to perturb the INPUT to make y-hat exactly correct. Forward error answers 'how wrong is my answer?'; backward error answers 'whose exact answer is it?'. The two are linked by the master rule: forward error is at most the condition number times the backward error.
The subtlety that defines this whole field: a large forward error is not automatically the algorithm's fault. If the problem is ill-conditioned, even a flawless (backward-stable) algorithm can produce a large forward error, because the condition number multiplies a tiny backward error into a big output error. So forward error alone cannot tell you whether to blame the method or the problem; you must look at backward error and conditioning separately to apportion the blame correctly.
Solve a system whose true x = (1, 1) and your code returns x-hat = (1.03, 0.97). The relative forward error in the 2-norm is about ||(0.03, -0.03)|| / ||(1,1)|| = 0.0424/1.414 ~ 3% — but whether that 3% indicts your solver depends entirely on kappa(A).
The headline number — but meaningless to interpret without the condition number.
Forward error is usually impossible to compute exactly in practice (you would need the true answer). It is estimated, bounded via conditioning times backward error, or checked indirectly through the residual.