this looks like either salt of mold. Neither is good. The salt is being soaked up by the clay plot and so it being removed from the soil (and the plant). This can often lead to the death of some sensitive plants like African Violets. If it is mold it is a problem as indoors that promotes allergies and can affect the plants too (though molds that grow like this are what are termed saprotrophs and grown on dead stuff as opposed to those that grow into live plants that are called biotrophs and a nastier because they have the chemicals to kill healthy plants but healthy plants can deal with that if not stressed).

Anyway, take the plant out. Change the soil and discard it. Soak the pots in a mild vinegar water solution and scrub off the white stuff. Let them dry very well.

Glazed pots dont have this problem but they are not as nice to look at.

good luck