For a relay switch and spring of a particular size, you need the a certain amount of magnetic force to be generated by the coil to overcome the spring force and close the switch. You can get the same amount of magnetic force in two ways:
1. Make the coil have lots of turns and have low current (high resistance because more turns).
2. Make the coil have few turns but have lots of current (low resistance because fewer turns).
So by changing the number of turns in the relay coil, we can make the same relay work at high voltage or low voltage. Either way will generate the magnetic field strength that we need, except one requires high voltage low current and the other requires low voltage high current.
The magnetic field strength of the relay coil depends on the current. The only thing the resistance of the relay affects is what voltage you can directly apply to it to get that current. If the relay's coil resistance is too low and your voltage is too much current flows making the magnetic field strength even stronger which isn't a problem but the extra heat might burn out the coil. In that case you can just add a series resistor in if you want to run it off a higher voltage. If your supply voltage is too low and the coil resistance is too high to get the required operating current, there's nothing you can do except to get a higher supply voltage or a lower coil resistance.
Have a question about Relays:
If high resistance relays draw less current then why cant we use it for all applications? Why do we require Low resistance relays?
So why do we require low resistance relays? THe answer is simple- so we have relays that work at lower voltages. A relay coil with a resistance that requires 1000V uses even less current. Why don't we use those everywhere? Well, not everyone has 1000V available. Different applications have different voltages available to them.
THen you might ask, well why don't we use low resistance relays for everything since we can use them with higher voltages just by putting a resistor in series with the relay coil? Well the answer there is that wastes power (sometimes a lot of power if the supply voltage is much higher than it needs to be). The power being dissipated is not being used to generate the magnetic field in the coil that closes the relay. Another reason is big heavy relays needs lots of magnetic force which means a lot of power. We can use lots of current at low voltage to get that high power, but because of I^2R it makes a lot more heat than if we used a higher voltage with less current.
As for what relay you can use, we need to know both the relay coil resistance AND operating current to know what voltage you can use. It's a simple V=IR problem. For example,
If your coil needs 1A to run, and has a resistance of 10ohms, then you need V=IR=(1A)(10ohms) = 10V. If the voltage you have is lower than 10V, you need more voltage or a new relay. If you have a voltage higher than 10V, you can get a relay that at 10V or you can put a resistor in series with the coil to limit the current in the coil to what you need so the coil doesn't burn out.