I have a 1858 new army replica, love it.  Problem is when I fire it, the other cylinders caps come off the nipples.  What am I doing wrong?

You are probably using #11 caps and need #10 or some times you need to pinch the cap a little to make it grip better.

HTH
Bob

ETA:  Or you could make a custom cap seizer by taking an old pair of pliers and drilling a hole with the jaws clamped in a vise, that is a  thousandth or so of an inch smaller than where the cap grips the nipple.  Than using the pliers with the newly drilled hole between the jaws,  size each of your caps to grip tightly on the nipples of your revolver.

Wear safety glasses when doing this though!

Yep. Improper sized caps on the nipples of the revolver.
Easy fix.

When you cap the nipple use a pair of needlenose pliers to SLIGHTY squeeze the cap mouth out of round. Then place on the nipple. If too loose bend the cap a little more. Trial and error.
or
Buy a different brand and size of caps. You'll just have to experiment on which brand/size works best on your revolver.
Most common is #11 size. I've found these to be unsatisfactory on my C&B's.
Most places sell CCI #11's. Look for Remington #10's. CCI #10's might work too.
I've found the Winchester Magnum #11's work okay on my C&B's.

If it's a Pietta, likely the nipples are improperly shaped (too thick is common). #10 caps won't solve that,as they're typically smaller than the more commonly found #11s.
I had to reduce the diameter on my Pietta's nipples to allow them to fit #11 caps properly. Even pinched caps would fall off of that damn thing,jamming the cylinder.
Now the caps stay on just fine,and split nicely when fired,making removal relatively simple.

Try Treso nipples:

Treso

They are very consistent and more durable than stock nipples.