What you need is a relay or contactor. The contacts are rated for the voltage and current of the power supply, e.g. 240V @ 10A. Relays are likely to be available for up to 10A, greater than that and it becomes a contactor. A relay with 240V contacts could operate the coil of a contactor, as contactors often have 240V coils.
The coil of the relay operates from a separate supply, but you want one with a DC supply of 12V or 24V. This coil is isolated from the mains, and you have a power supply just for it. The coil itself is operated by a relay driver circuit (which normally inverts the logic). The input to that is logic level.
It may be that a solid state relay will be suitable (second link). These are operated by logic levels directly and come in 240V x current ratings at least 10A. They work well with resistive loads, but in my experience not so great with inductive or reactive loads.
The first link below shows a typical relay driver. This will not suit all situations, as it depends on the relay coil current. The transistor could be a 2N2222. This will work up to 0.5A or more, but the 10K resistor should be reduced so the base current is 10mA or more (270 ohms), which in turn depends what logic circuit is driving this. The transistor could be a 5 or 10A logic level power mosfet instead (N channel). This will work with any logic and drive even more current.