Both forms are correct. Either weather or wether are correct forms, however, they are two different words. Things that connect these two are very similar spelling and the same pronunciation, they both sound like that – /ˈweð.ər/. Both words can act as nouns, but weather in some contexts can also be a verb. The main distinctiveness is a completely different meaning.

Correct spelling, explanation: the conditions in the atmosphere (noun) or to deal with a problem or a difficult situation (verb).

Correct spelling, explanation: (used rarely) a male sheep that has been castrated.

Definition of weather:
Noun: the conditions in the atmosphere such as wind, temperature, cloudiness, etc. visible in a particular area;
The weather forecast for the next week predicts heavy rain and temperatures below 10 degrees Celsius.
Verb: to deal with a problem or a difficult situation;
She weathered the complicated situation with her studies and got the diploma in September.
– to change visually over a period of time because of the air conditions;
Our fence weathered so badly during winter, we have to repaint it soon.

Definition of wether:
Noun: a male sheep that has been castrated;
This sweater from wether lambswool is so delicate, but also incredibly expensive.