The word kille can refer to a young man or to someone's boyfriend – you can think of it as the male equivalent to the word tjej.
You can listen to how the word is pronounced below.
There's no hard and fast rule dictating when a boy stops being a pojke and becomes a kille, or when a kille becomes a man, and there are definite overlaps in terms of age. Generally, you can use it from the teenage years and to describe men in their 20s and 30s.
Context plays a part too; kille started out as a slang word and is a more informal term than man, similar to 'guy' in English, or 'bloke' in British English. However, it's becoming more and more commonly used, and unlike 'guy' or 'bloke' you'll see it in some formal contexts, such as news articles. This is probably because newspapers like brevity, and using kille instead of man gets extra information about the person's age across without using extra words.
When used with a possessive pronoun, it means 'boyfriend', so a person referring to min kille is talking about their partner. Note the difference between jag ska träffa en kille i kväll (I'm meeting a guy for a date tonight) and jag ska träffa min kille i kväll (I'm meeting my boyfriend tonight). In the first example, there's no existing relationship between the two.
And it pops up in a few compound nouns: killkväll (guys' night), killkläder (men's clothes, almost always targeting the younger demographic), and killkompis (guy friend), for example.
As for where this word comes from, it's linked to the English word 'kid', which has referred to young goats and has been an informal word for young children for several centuries. Both English 'kid' and Swedish kille come from the Old Norse word kið, which meant 'young goat', and the diminutive suffix -ling was later added to emphasize the 'young'.
In today's Swedish, killing is the word for 'kid' meaning 'baby goat', while kid means a young deer (although it was earlier used to refer to any baby animals), and kille means a young male human.
Kille is also the name of one of Sweden's oldest card games, but most of the time, you can assume it means 'guy'.
Jag och min kille har varit tillsammans i tre år
My boyfriend and I have been together for three years
En kille blev rånad på tisdagskvällen
A young man was robbed on Tuesday evening