question for green thumbs

hi there, those who tend plants.

i like plants a lot. over the past year, i've noticed a kind of white spore growing on the leaves and stems of two of my favorite plants. the spores don't seem to affect the overall health of the plant, but it looks kind of gross and has got me worried. if you touch the spores, they're a kind of mushy. i've tried washing the plant off but i can never get the spores out altogether and they come right back.

does anyone know what this is? should i just trash the plant? do the spores threaten my health or that of my other plants?

i suppose i should be asking this in some kind of gardening forum but (a) i always like reading ilxor responses and (b) they'd probably make fun of my ignorance.