As multiple people here have correctly pointed out, the amp rating is a MAXIMUM rating, just the 5V is a "fixed" property of the supply.
So in theory, you can use a 5V 500mA supply just as well as a 5V 1000A supply to supply your project with 250mA.
If you exceed the rating, in the best case, the supply will shut off, in a slightly worse case, the voltage will drop (resulting in unexpected behavior), in the worst case, it will get hot and catch fire.
In some conditions, it might be interesting to look at the efficiency and the quiescent current. I have seen phone chargers that draw 2W to power an ESP8366 of 0.5W. This might not be important for playing around, but if you have 20 devices lying around always on for home automation, than it gets interesting. A current rating that is closer to the actual demand has the tendency to have better efficiency, but in the phone charger range, there is also a good bunch of quality involved.