My understanding is that for each tradable asset, there are two prices: Bids and asks; and trades will only happen when asks and bids exist at the same price point.
Most prices we see on tickers are the lowest asks, I believe.
Most commonly, price you see on ticker quotes is the price the stock last traded at; i.e., the last price at which a buyer and seller exchange money for equity.
Yes, a trade represents an instant in time when a potential buyer and a potential seller agree on a price. The highest bid matches the lowest ask, and then the exchange "matching engine" generates a trade from them.
Instantaneously afterward, the highest remaining bid will be below the lowest ask.
When anyone talks about "the price" of a stock it is an approximation of some sort. Probably they mean the last traded price; maybe the midpoint (bid+ask)/2; maybe the symbol is in the middle of an auction and they mean the auction clearing price.
There's an order book listing buy and sell orders at a variety of prices clustered around big/ask, but the actual bid/ask numbers shown are a standardized summary view of the order book