The current rating of a power supply is the maximum it can deliver if the load demands it. A power supply can not dictate both the voltage and the current. In this case the supply will keep the voltage at 5V and the load will draw whatever it needs. The 1A rating means the load can draw up to 1A before the supply might not be able to keep its output at 5V.
That all said, there are some supplies that are designed to require a minimum current to function. Those that work like that usually require something like 10% of the maximum rated current. In this example, it would mean the supply is only promising to keep the output at 5V if you draw from 100mA to 1A. What happens above or below that depends on the supply. The only way to know if your supply has a minimum current requirement is to check its datasheet.
If your board draws 250mA, then almost certainly the supply will be able to maintain the specified voltage, but always check the datasheet anyway.
In no case can the supply somehow force 1A thru your board at 5V if the board only wants to draw 250mA.