Stocks prices are determined whenever a buyer and seller agree to trade at a given price. The company (you use AAPL as an example) doesn't set its own stock price. Rather, the investors set the price every time it trades. There's no "official" price -- just the last trade.
Likewise, you can offer to trade a stock at whatever price you want: that's the definition of a limit order. You might not find a willing buyer or seller at that price, but you can certainly open an order.
Stock quotes that you get from your broker or a finance web site reflect the price as last traded. These quotes are updated throughout the trading day and the frequency and delay varies amongst quote providers.
Like Knuckle-Dragger suggests in the comments, there are ways to get real-time quotes.
It's often more helpful to think in terms of bid/ask instead of "official price". See this question for details.