Uncirculated coins (coins that show no signs of wear) are worth more than circulated coins (coins that have been used in everyday transactions and show some wear). Coins with certain errors or special characteristics, such as double strikes or over mint marks, can also command higher prices.
The best way to determine the value of a particular 1903 ten dollar gold coin is to consult a professional numismatist (coin expert) or to research online coin price guides.