the counting process N(t)
Imagine a turnstile clicker that ticks up by one every time someone walks through, and you record the running total as the day goes on. At any moment t, the number on the clicker is N(t): how many events have happened from time 0 up to time t. As time moves forward the count only ever goes up, in unit jumps, never down — a staircase that climbs one step at each event and stays flat in between.
Formally, N(t) is a stochastic process indexed by time, taking values 0, 1, 2, ... It starts at N(0) = 0, is non-decreasing, increases by exactly 1 at each event (jumps of size one, no ties), and is right-continuous (at the instant of an arrival it has already counted it). The natural companion to counting is the increment N(t) - N(s), the number of events strictly between two times s and t; for a Poisson process this increment is Poisson with mean lambda*(t - s) and is independent of what happened before time s. The counting view and the interarrival view are two readings of the same picture: the n-th jump of N happens exactly at the n-th arrival time.
Why bother with N(t) rather than just listing arrival times? Because many questions are about counts: 'how many claims this quarter?', 'is N(t) above capacity?', 'what is E[N(t)]?' For the Poisson process E[N(t)] = lambda*t, a straight line through the origin. More generally, ANY model of arrivals can be described by its counting process, so N(t) is the common language for Poisson processes, renewal processes, and birth-death processes alike — it is the object you study when you care about totals over time rather than individual gaps.
Events arrive at times 1.2, 3.8, 4.1, 9.0. Then N(t) = 0 for t < 1.2, jumps to 1 at 1.2, to 2 at 3.8, to 3 at 4.1, to 4 at 9.0. So N(5) = 3 (three arrivals by time 5) and N(9) = 4. The increment N(9) - N(5) = 1 counts the single arrival in (5, 9].
N(t) is a right-continuous staircase: it jumps by 1 at each arrival and is flat between.
N(t) counts events in [0, t], so N(t) and the arrival times carry the same information — but N(t) is the right tool when you want totals (E[N(t)] = lambda*t) rather than individual gaps.