sequence
Imagine an unending line of numbered seats, with exactly one number assigned to each seat: the first, the second, the third, and so on forever. A sequence is precisely this — an ordered, never-ending list of values, one for each natural number. Unlike a finite list you could write out in full, a sequence has a first term but no last term.
Formally, a sequence of real numbers is a function a from the natural numbers to the reals, written a: N -> R. We usually denote the value at index n by a_n rather than a(n), and write the whole sequence as (a_n) or (a_1, a_2, a_3, ...). The index n is just a label recording position; the order matters, and repetition is allowed, so a sequence is not the same as the set of its values.
The whole study of analysis begins here, because nearly every limiting process — convergence of series, continuity, derivatives, integrals — can be phrased in terms of what a suitable sequence does as its index runs off to infinity. Do not confuse a sequence (an, ordered, infinite list) with a series (a sum of such a list); they are different objects built from the same raw material.
The sequence a_n = 1/n is (1, 1/2, 1/3, 1/4, ...); the sequence b_n = (-1)^n is (-1, 1, -1, 1, ...). Both are perfectly good sequences, even though only the first one settles down.
A sequence may converge, oscillate, or grow without bound — being a sequence says nothing yet about its fate.
Indexing often starts at n = 1, but starting at n = 0 (or any fixed integer) is equally common; it changes nothing about convergence, since only the eventual behavior matters.