I have a project where I have a 25V power source powering elecronics that I want to be able to switch on and off using an arduino pin. What would be the simplest way to effectively have a digitally controlled on off switch given the differences in voltage? Sorry I am new to this and appreciate any help 
How much current does the 25V circuit carry?
only about 65 mA
The simplest way is with a relay. A relay is an electrically-isolated and electrically-operated switch. At 24V and 0.65A almost any relay will work, but it needs a 5V coil (so it can be switched with 5V).
The one "catch" is that the Arduino can't directly drive a relay coil so you need a driver circuit. But you can buy a "relay board" with a relay and driver built-in.
Or, there are solid state relays that don't need a driver. The tricky thing with solid state relays is that AC & DC relays are not interchangeable so assuming you have 24VDC, you'd need a DC relay (that's specified for 5V control).
DVDdoug:
The simplest way is with a relay. A relay is an electrically-isolated and electrically-operated switch. At 24V and 0.65A almost any relay will work, but it needs a 5V coil (so it can be switched with 5V).The one "catch" is that the Arduino can't directly drive a relay coil so you need a driver circuit. But you can buy a "relay board" with a relay and driver built-in.
How is that simpler than just using a transistor?
--- bill
I suppose because a transistor requires some knowledge regarding the voltage difference and type whereas a 5v relay board will pretty much just work for someone with low electronics knowledge with less chance of letting the smoke out
Just pick of a simply relay like this You can find them cheaper on ebay, banggood, etc. There are many to choose from but all use the SRD-05VDC-SL-C relay.
use an optocoupler with 220 ohm in series on input, its like turning an led on, where the output handles the 24V circuit i think a little bugger like 4n35 should be able to handle 100mA on its own, simple, cheap, small, very safe due to galvanic insulation, silent and reasonably fast, less power needed then a relay, no inductivity...
for greater power applications switch a gate or base with it,
for ac you can use an optotriac-triac combination
optos are genius little things XD