I frequently shrink down large images for online photo galleries. What I want to know is, when I tell photoshop, et al, to shrink my image to a certain size/percentage, will it retain better quality if I use round percentages like 25%, 50%, etc, or does it make no difference?
There was a time (10-15 years ago) when sticking to specific percentages was helpful. Often 125% or 75% were best. You would simply repeat enlargements or reductions at this percentage until you got close to the size you wanted then the final step would match the desired final size. This resulted in sharper, clearer, pixels since less interpolation was needed for each step. This was done due to the limitations of software. Specifically, Photoshop seemed to interpolate files better at some percentages. This may be why you've heard or read of this practice before.
That being posted, this practice is no longer needed. Software has advanced and it's no longer necessary to massage any interpolation to achieve better results. As Bakabaka points out in his answer, using the Bicubic Automatic setting will do a great job in practically every instance regardless of the percentage you use. I've, personally, not felt the need to step-enlarge/reduce in a number of years.
When you think about things which are available today, such as content aware fill, it may be clear just how far the software has come and why the step interpolation practice is a thing of the past.
The only exception would be using Nearest Neighbor in some instances to keep pixel edges sharp. And, if using Nearest Neighbor, it can still be helpful to do a couple small step interpolations rather than one huge interpolation, but there's no specific percentages which yield better results (other than whole percentages).