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Let's say we want to buy stock X at $1

We say we will buy at $1, but by the time our request arrives to the seller/transaction broker, the price has moved to $1.05

Does our request just get denied or do buyers get to say we want to buy stock X between $1 and $1.50?

This way, there's a buffer to allow for some stock price movement while our transaction is in the pipe. Is this how it works?

If it is, is the buying FIFO (First in first out) or do large buyers/sellers get to jump to the top of the queue?

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The buying and selling is a transaction with stocks as with anything else. For every buyer there's a seller, and if there's a selling order to match the buying order - the transaction will go through. When you're giving a "Market" order, the broker will look for the next selling order to match to your buying order, and the price requested by the seller would be the price you pay. If it's a limit order, then you will be matched with any selling order within the given limit, and if by the end of the period no such order is found - there'll be no transaction.

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  • Importantly, you get the best price for your transaction. If you're buying, and someone is offering the stock for $2 and someone else for $3, you'll pay $3. This is the same for both market and limit orders.
    – Kartick Vaddadi
    Oct 30 '16 at 5:13
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I'm really not qualified to respond to the second part of your question. But you'll want to look into the difference between market orders and limit orders.

A market order is an order to buy or sell a security at the market price when the order is filled. A limit order is an order to buy or sell a security at the market price, given that the market price is "better" than a certain threshold.

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