the coupling at each vertex

When two people meet, how likely they are to strike up a conversation depends on how outgoing they are. In a Feynman diagram, every vertex is a meeting of particles, and how readily that meeting happens is set by a single number called the coupling. A weak coupling means the interaction is shy and rare; a strong coupling means it happens easily and often. The coupling is, in plain terms, the strength dial of a force at the level of a single interaction point.

Each vertex in a diagram contributes a factor equal to the coupling, and the probability of a process turns out to depend on the coupling squared (because probabilities come from squaring the amplitude, as the golden-rule intuition makes precise). So a diagram with more vertices carries more powers of the coupling and, if the coupling is small, contributes much less. For electromagnetism the relevant combination is the fine-structure constant, alpha, which is about 1/137 — a small number, which is exactly why each extra order — two more electromagnetic vertices — makes a diagram's contribution roughly a hundred-odd times smaller. For the strong force the coupling is larger, which is part of why strong-force calculations are so much harder.

Counting powers of the coupling is the heartbeat of how predictions get made. It tells you which diagrams matter most (fewest vertices, fewest powers of the coupling) and which are tiny corrections you can often ignore. A subtle but important fact is that the coupling is not truly a constant: its measured value drifts with the energy of the collision, an effect called the running of the coupling. So the dial setting depends on how hard you are probing.