Hi,
I am building a clock(simple, I know). In this particular build of the clock I need to independently control 26 LEDs. The maximum that will be on at a single moment is 25. What kind of power will I need, is the 5v enough?
Yes. Need up to 20mA each if all will be on at full brightness.
Do you plan to drive each one directly with an arduino output pin?
Or multiplex them somehow? One way is to use common cathode design, drive all segments high in parallel from arduino outputs, but only connect 1 digit's cathode to ground at a time using NPN transistor or ULN2003/2803 type part.
Or drive each digit with its own shift register, you update each register individually as needed.
Lot of options.
5V will be plenty of voltage as long as you aren't putting LED's in series.
If your LED's need 20 mA each your total current draw is 25 * 0.020 Amps = 0.5 Amps. That will be too much for running off USB but I think the built-in regulator can handle it if you provide 7 to 9 Volts to the power jack (or the Vin pin). Get a power supply that can provide at least one Amp.