I just noticed my atv I purchased used has tires sized 25x8-12 on front and 26x10-12 on the rear. According to the manual it should have 24x9-12 on front and 25x10-12. Is it ok to stay with these sizes? Will it cause any mechanical problems?

 

If it is 2 wheel drive there is no problem at all. If it is 4 wheel drive the ratio is off a little but should not bother any, just make sure not to drive it in 4 wheel drive on asphalt. If anything, it will cause premature tire wear, on extreme cases it may put some extra strain on the driveline. You should be fine.

 

It is a 4WD. So if I was to change them for new tires...what would be the best size? I can't seem to find any brand other the original brand tires that have the same front sized tire. Should I go with 24x8x12 for the front and 25x8x12 for the rear? Or can I go with 25x10x12 for the rear and leave a narrower tire on front? The secondary transfer gear was replaced a while back....you think incorrect tires may have helped?

 

The width should not make much difference so long as the tire is wider than the rim. I like it when the tire is at least 1 inch wider than the rim.
The circumference is the critical measurement. In your case since the factory had a 24 inch on the front and a 25 inch on the rear then putting a 25 inch on the front and 26 inch on the rear is ok (the ratio is only off 0.0016666 of an inch - absolutely nothing to worry about).
You can run what ever width you want on either front or rear. Most people have the rear tires 1 or 2 inches wider. Some run the same width on front and rear.
In your case 8 or 9 inches on the front is fine, 10 or 11 inches on the rear is fine, just so long as the rims are about 1 inch or more narrower than the tire.

24x8x12 means: (24) inches diameter of the tire, (8) inches tire width, (12) inches diameter of the wheel/rim. No wheel/rim width is specified on the tire size, in this case then any rim that is 5.5 to 7.5 inches wide is ok, preferably a rim 6 to 7 inches.