the rate (intensity) parameter

Every Poisson process has a single dial that sets how busy it is: the rate, written lambda. If buses pass your stop at lambda = 4 per hour, that one number captures everything about the speed of the process — turn it up and events come thicker, turn it down and they thin out. It is the long-run average count per unit of time (or per unit of space, in the spatial version).

Precisely, lambda is the expected number of events per unit time, so over a window of length t you expect lambda*t events, and indeed the count in that window has mean lambda*t and (because the Poisson distribution has equal mean and variance) variance lambda*t too. It controls the gaps as well: the time between consecutive events is exponential with rate lambda, so the average gap is 1/lambda. A higher lambda means shorter typical gaps and more events; a lower lambda means longer waits. The units must match the question — '4 per hour' and '1/15 per minute' describe the same process.

Two honest points. First, lambda is a rate, not a probability: it can exceed 1 (e.g. 100 packets per second) because it is events per unit time, not a chance of something. Second, the plain Poisson process assumes lambda is constant; when arrivals speed up and slow down (lunchtime rushes, daylight) the rate becomes a function lambda(t) and you have a non-homogeneous process. Estimating lambda from data is usually as simple as counting events and dividing by the total observation time.