The concept of infinity (∞) is perhaps the most important new idea in calculus. Specifically, calculus is about procedures with infinite number of steps, or infinitely small steps.
The derivative is a slope calculation (rise/run) with an infinitely short run. The integral is a rectangles-approximation-to-an-area using infinitely thin rectangles, and series are summation procedures with infinite number of steps.
High school math deals with procedures with finite number of steps, whereas in calculus we learn to use infinity as part of our calculations. The reason why limits are important is because they allow us to make certain statements that would otherwise not be true:
1/n ≠ 0 even if n is huge
but lim_{n -> ∞} 1/n = 0
sum([1/2^n for n in range(0,N)]) = 1.999999... ≠ 2
but sum([1/2^n for n in range(0,∞)]) = 2
(examples taken from my math book)