Kärring is another way to say 'woman', but beware: this word is usually seen as derogatory and is often translated as 'hag', so be very wary as a non-native speaker.
It's reserved for older women only, and has connotations of foolishness and being unkempt as well, so the phrase som en kärring (like an old woman) is often used as an insult, for example hon klär sig som en kärring (she dresses like an old woman). It's also frequently combined with negative adjectives, such as galen kärring (crazy old woman), or some even less polite alternatives.
However, that's not always the case. Sometimes kärring is used as a term of endearment, usually by an older man about his wife, and this is probably due to a misconception about the origin of the word.
Kärring is sometimes spelled with only one 'r', which gives a clue into its history.
It comes from kärling, an older word for 'woman' related to karl (man), which still exists in Swedish today. Over time, the middle consonant cluster 'rl' got eroded down to 'rr' or 'r', a process which is called fricativisation or rhotacism in linguistics.
So kärring has nothing to do with the word kär ('in love' or 'dear'), but many Swedes think it does, which might be why many older men use it as a term of affection for their partner. While karl was a Germanic word that can be traced back for centuries (and yes, this is the origin of the names Carl and Karl, which literally mean 'man'), kär comes from Latin carus (beloved), and is related to French cher and Italian caro as well as Norwegian kjær.
As for why kärring has become loaded with such negative connotations, despite the neutral original meaning of kärling, that's probably due partly to a negative societal attitude towards older woman, and partly because the Swedish language had alternative neutral words for 'older woman', like gumma, so kärring's meaning became more specific.