bounded operator
In finite dimensions every linear map is automatically continuous, so you never had to think about it. Infinite dimensions break that gift: a linear operator can stretch some directions infinitely more than others and tear continuity apart. A bounded operator is one that does not do this — it magnifies lengths by at most a fixed factor, no matter which vector you feed it. Bounded is the right word for a well-behaved linear map between normed spaces.
Precisely: a linear operator T from X to Y between normed spaces is bounded if there is a constant C with ||T x|| <= C ||x|| for all x. The smallest such C is the operator norm ||T||. The fundamental fact is that for linear operators, bounded, continuous, and continuous-at-0 are all equivalent — a single inequality controls the whole map. So boundedness is exactly continuity, just phrased quantitatively.
Why this is the correct notion of map in infinite dimensions: unbounded linear operators exist and are everywhere (differentiation is the headline example), but they cannot be continuous, cannot be defined on the whole space in any usable way, and behave wildly. Restricting attention to bounded operators recovers a clean theory. The set of bounded operators from X to Y is itself a normed space, and when Y is complete it is a Banach space.
A caution: do not confuse bounded operator with bounded function. A bounded operator is almost never bounded as a function — feed it bigger vectors and ||T x|| grows without limit. Bounded here means the ratio ||T x|| / ||x|| stays bounded, i.e. the operator has a finite amplification factor.
One uniform inequality is equivalent to continuity of the whole linear map.
Slogan: in infinite dimensions, bounded = continuous for linear maps. The whole subject of operator theory is largely the study of bounded operators, with unbounded ones handled by careful domain bookkeeping.