Whether, weather and wether are homophones, i.e. words that sound the same, but have different spellings and meanings. That is to say…
Whether:
i. A conjunction used to express doubt, or a choice between alternatives, as in “They seemed undecided whether to go or stay.”
ii. To express an enquiry, as in “I’ll check whether the item is in the case.”
Weather: the state of the atmosphere in terms of temperature, wind, cloud, and precipitation.
Wether: a castrated ram (a male sheep) or a castrated billy (a male goat).
So don’t rely on an automatic spellchecker to do your proofreading!