Everything else in trading is commentary; the matching engine is where a trade actually happens. It receives messages in the order they arrive at its gateway, applies the venue's priority rules, and emits fills and book updates.
Two consequences follow. First, arrival order at the engine — not the time you clicked — decides who gets filled, which is why colocation and latency are worth money. Second, the engine is deterministic: given the same message sequence, the same trades happen, which is what makes queue-position a calculable thing rather than a guess.
Example: two traders send buy limits at 25.00 within 80 microseconds of each other. The one whose message reaches the gateway first is ahead in the queue for every share sold at 25.00 that day. If only 4,000 shares trade at 25.00 and the first order is for 5,000, the second trader gets nothing.
Related: price-time-priority, queue-position, central-limit-order-book, colocation