Jinx appeared for the very first time during Pool Party 2013 (Her head can be barely seen next to Pool Party Renekton [S|L] in Pool Party Graves [S|L]' artwork, and the corresponding login screen at the time was drawn from her first-person view).
Jinx straightforwardly means "ill omen", from the Pre-Greek noun ἴυγξ iynx "Eurasian wryneck", a bird often used as a charm..
Jinx is a Zaunite criminal who is wreaking havoc across Piltover 'for fun' and to provoke Vi.
Jinx and Vi are often shipped by the community as sisters. The only reference to this on Jinx's release was her quote,"You think I'm crazy? You should see my sister", but this sister is likely a figment of her crazy and not a real person. In any event, there was never any implication that sister was Vi, but players equated Jinx's antagonization of Vi specifically as "sisterly bonding".
Incidentally, Jinx only antagonized Vi specifically on English-speaking serves. Other servers showed her antagonizing other champions, such as Lucian.
Shopkeeper Lyte later says to Vi, "You look like your sister! Er, wait, I'm not supposed to talk about that... ". This could be a confirmation that Vi (at least) has a sister, but could also be an example of Lyte repeating rumours he's heard from players (such as the rumour he heard from players that Lux and Ezreal were dating, which Lux herself later denied after hearing the rumour herself).
There is a significant indication that the pair did grow up together, however. With Vi currently suffering from amnesia, Jinx has taken it upon herself to try and trigger her memories.
In an interview with Ghostcrawler, he mentioned that Jinx and Vi are "sisters", but later confirmed he was merely stating what he thought was true, and that the narrative team said otherwise.
Ekko remembers Jinx from before she went mad ("I had a crush... until you started talking to the gun")