Folkkär is a pretty word with a positive meaning. It refers to a person (usually) or a thing which is well-known and loved by people.
You could translate it as 'widely loved' 'beloved by the people', or 'a popular household name', and it can easily be broken down into two parts: folk ('people' or 'population') and kär ('in love' or 'dear').
It's an adjective, so you might say 'efter att han vann Melodifestivalen blev han riktigt folkkär' (after he won Melodifestivalen he became a popular household name) or 'den folkkäre journalisten kommer att skriva en ny bok' (the widely loved journalist will write a new book).
As you might have gathered, it's different from the more neutral word kändis ('celebrity') which simply refers to someone who is widely known, no matter what public opinion of them might be.
And the use of folk is interesting too. Swedish has several other words for 'popular', including populär or the abbreviation poppis, and omtyckt.
As word nerds might already know, the words 'popular' and populär come from the Latin term populus (people) and literally mean 'liked by the general public'. Folk generally refers to 'people' in the sense of a national population, so a folkkär singer, athlete or author would generally be one who is considered a national treasure. They're not just popular, but they mean something to the country and its culture. So while a populär or omtyckt person might be well-liked by their immediate community, friends or colleagues, someone who is folkkär is known and loved across the country.
In Sweden, you're more likely to hear folkkär used to talk about homegrown talent or figures who are particularly highly esteemed within Sweden, rather than international celebrities.
And because of the personal tone of the word kär, it generally refers to those celebs who you feel like you know and would be down-to-earth if you ever met, rather than those who are all too aware of their fame and talent (for this reason, it's debatable whether household name and footballer Zlatan could be described as folkkär).
Good examples would include Tilde de Paula Eby, a TV presenter who has fronted TV shows from news programmes to Swedish Bake-off and Let's Dance, or the late artist Lill Babs whose six-decade career included singing and acting.
By default, folkkär tends to refer to people who are popular in Sweden specifically, but you can also say things like 'artisten är väldigt folkkär i Tyskland' (the artist is widely beloved in Germany) or 'hon är en av Storbrittaniens folkkäraste författare' (she is one of the UK's best loved authors).
Sara Danius var doldisen som blev folkkär
Sara Danius was the nobody who became a beloved household name (Expressen)
Hans inspirerande filmer har gjört honom folkkär
His inspiring films have made him widely popular