Oväder is a useful Swedish term without a direct one-word equivalent in English. You could translate it as 'bad weather', 'inclement weather', or even 'storm', depending on the context.
On its own, the noun väder means 'weather', which comes from the Old Swedish term veðr and is related to the equivalent words in many other languages: English 'weather', German Wetter, Danish vejr, and Dutch weer, to name a few.
In older forms of most of these languages, including Swedish, väder was used specifically to refer to windy and/or rainy weather, but over time became generalized to refer to the condition whether good or bad.
But traces of the original meaning remain in Swedish in words such as väderkvarn (windmill), and the colloquial expression munväder (literally 'mouth wind', meaning 'nonsense'), as well as place names such as Väderstad (literally 'windy town'). Väder might also be used to refer to flatulence, with the expression släppa väder literally meaning 'to break wind'.
So what about that o?
The prefix o is most often used in Swedish as a form of negation for adjectives, adverbs, verbs and nouns, similar to the English prefix 'un-'. For example, rimlig (reasonable) becomes orimlig (unreasonable), lycklig (happy) becomes olycklig (unhappy), en möjlighet (a possibility) becomes en omöjlighet (an impossibility), and gjord (done) becomes ogjord (undone or unfinished).
The prefix o as a negation is especially common in the Norrland region, where it's used in many more instances than it is in the rest of the country.
For example, in Norrland speech, o can for example be added to adjectives such as bra to create obra (literally 'not good', but most Swedes would say dålig, 'bad', or simply inte bra). It can also be combined with many more verbs than in standard Swedish, where inte would be used, for example: jag har osovit (I haven't slept), jag har okommit fram (I haven't arrived) and so on.
In standard Swedish, o has several other nuances beyond direct negation.
It can also be used as a prefix to denote an abnormal, extreme form of the root noun, implying judgment from the speaker. For example, the word odjur means 'beast' or 'monster', from o + djur (animal), while ogräs (o + gräs, literally 'un-grass') means 'weed', referring to undesirable plants, and ovän (o + vän, 'friend') doesn't simply mean someone who is not your friend, but someone who is an enemy.
In these cases, the o isn't a negation but a sign of abnormality and undesirability; a distorted form of the root word. And oväder falls into this category: it doesn't refer to a lack of weather but rather weather which is unusual and extreme, with negative consequences. Think of it as weather which is so severe and unpleasant, it isn't worthy of the name 'weather'.
As for when you can use it, oväder is quite versatile. It can be used in place of English 'storm', although Swedish also uses the word storm, as well as åskväder (literally 'thunder weather') to refer to thunderstorms. There should be some form of precipitation to classify as oväder, either snow, rain, or hail, and it usually indicates winds as well, but there is no official meteorological definition.
You can use it as a countable noun (ett oväder, två oväder) to refer to specific instances of precipitation or storms, for example ett tillfälligt oväder (a temporary storm) or as an uncountable noun for more general descriptions, such as vilket oväder vi har fått (what bad weather we've had).
Inställda tåg på grund av oväder
Cancelled trains due to extreme weather/storms
Nytt oväder på väg in över hela Sverige
Another storm/instance of extreme weather is on its way across all of Sweden