Finite and infinite

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In mathematics, the meaning of the terms finite and infinite varies according to context.

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Essentially, finite means (similar to common usage) having a size which is bounded by a natural (or, equivalently, by a real) number.
while infinite means unbounded in size or, more precisely, exceeding all natural (or real) numbers in size.
(Often bounded and unbounded is used in the same sense.)

But "size" may mean length, area, or the result of any other measurement, and thus the precise meaning of "finite" varies accordingly, but is often not explicitly given. Examples are:
finite interval, finite value, finite integral, finite degree, finite dimension, finitely often, etc.

A special case of size is cardinality, i.e., size with respect to the number of elements:
finite sets have finitely many elements, i.e., 0 or 1 or 2 or 3 ... elements,
infinite sets have more (i.e., at least an unlimited sequence of) elements.

Thus the interval of real numbers between 0 and 1 is a finite interval and a bounded set because its length is bounded, but it is an infinite set because it contains infinitely many numbers.