What causes white, fluffy stuff to build up on the inside of a brick basement wall?

What you’ve noticed on your basement wall is caused by a slow, steady migration of moisture through the brick from surrounding soil, bringing minerals as it does. In most cases you never actually see liquid water, since the inward migration is slow enough that the water evaporates as fast as it arrives. As this inward movement and evaporation happens, minerals are left behind on the brick, forming the deposits you’ve seen. They’re not harmful, so there’s no need to worry. The only way to stop them is with a coating that stops moisture movement. Lots of products claim to do this, but few actually deliver. Xypex and Drylok are two that I know work well.