
You might be here because you searched "what are brown fuzzy balls on cactus." Fear not, this is what a cactus bud looks like!
Fuzzy doesn't always mean soft though, be careful handling buds since some varieties have poky bristles and spines in with the hair. The example at left is from a torch cactus (trichocereus x echinopsis) which bud prolifically. A lot of cacti form a hairy brown of yellow bud similar to this before blooming.
If your cactus produces this type of bud, don't hold your breath waiting for it to bloom. The buds lay in wait on the cactus for quite a while until conditions are right, then all of the buds on the plant will often bloom simultaneously—or take turns blooming in a cadence like popcorn popping off. My torch cacti respond to the start of the monsoon season in the spring, having their most impressive blooms when humidity rises and Tucson gets its first rainstorms.
You'll know the tribble buds have become 'activated' when they grow fatter and elongated, especially when you start to see shiny fresh flesh reflecting between the hair. It will take several days to develop a larger bud on a torch cactus, like this one a day before its bloom:

Okay but why has this fuzzy bud been on my cactus for years?
It ded.
Sometimes buds abort. A lot of cacti and succulents can abort buds if moved significantly, repotted, or otherwise stressed. This is also a good reason to dead-head your cacti, since prolific bloomers like torch cacti will abort buds once they start to grow seeds. Even a well-cared for cactus will drop a certain percentage of its buds; I end up with a drift of cactus tribbles every year.
After a while you will learn the difference in color between live and dead buds on your individual plants. Until then use a pair of tongs to pick the buds from between spines: fresh buds will be firmly attached, while aborted buds are loose. If you look closely you may see buds that are hovering just off the surface, stuck in the spines.
