This is best explained through an example.
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Let's say its 6pm ET, the market is closed, and Google ended the previous day at $100/share.
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Moe enters a market order in dollars for $100. Leila enters a market order in shares for 1 share.
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The next morning, the stock price opens at $110/share (the US markets have after-hours and pre-market trading sessions that can affect the share price mostly available to institutional investors). Both Moe and Leila's trades are executed at 9:30am ET.
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Moe gets 0.909091 shares for his $100.
Leila gets 1 share for her $110.
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That's the difference between the 2 market order types!