Disdain can be either a noun or a verb. As a noun disdain means the general attitude of something or someone being beneath consideration or not valuable enough for respect.
The verb form is transitive, or used with an object. To disdain something is to mock it or judge it poorly. One can also disdain an action, in other words, refuse to complete the action because one has deemed it beneath consideration or unworthy.
The adjective form is disdainful, which describes something or someone has having a lack of regard or respect. Derivatives of this form include the adverb disdainfully and the rare noun form disdainfulness.
Distain is an archaic word not listed in all dictionaries. It meant for something to be stained or disgraced. However, generally it is most often seen in publications as a misspelling of disdain.
The main confusion between these two spellings is that the d sound and t sound are almost identical in formation and their only difference is that one is voiced and the other is unvoiced. Add in regional dialects and it would be hard to know how to spell this word based on hearing it alone.
Examples
It’s worth noting that most Democratic voters do not share the disdain that party leaders and grass-roots activists display toward expanded international trade. [The Wall Street Journal]
This was not just idle abuse, however: Healy knew that saying farewell to opponents in such a disdainful way would help undermine their confidence further. [Australian Times]