You can find everything you ever wanted to know about Jate here. To (very) quickly sum them up, they met just after the plane crashed and had an instant connection. However, Jack soon found out that Kate was a fugitive and began not to trust her... and then there was Sawyer. Kate was very torn between the two men, each with very different personalities.

When Jack, Kate, and Sawyer were captured by the Others, Kate slept with Sawyer because she tought he was going to die. Jack found out. Jack and Juliet (a reformed Other), much to Kate's dismay, got very "friendly".

But at the end of season three (just after Jack kisses Juliet), he turns to Kate and tells her he loves her.

Both are rescused as part of the Oceanic Six. Sawyer jumped out of the helicopter to save them, but before he did, he asked Kate to do him a favour and kissed her goodbye. While Kate's feelings were always in question, Sawyer's weren't - he loved her.

Off the Island, Jack and Kate are happily playing house with their "son", Aaron. They even get engaged. However, Kate is still trying to keep her promise to Sawyer, without Jack knowing. Jack finds out, and they fight.

Jack soon realizes that they have to go back to the Island, and nearly goes mad trying. He finds himself addicted to pain killers and an alcoholic... and without Kate.

Once he convinces her to go back with him, she gives up Aaron, and is distraught. She runs back to Jack and spends the night with him. Then, along with the rest of the Oceanic Six (minus Aaron), they go back to the Island.

Their relationship is still rocky, and Jack realizes that Kate may have never liked him, while she tried to convince him that "it was not all misery."

In the season five finale, as Jack tried to "reset time" so that they can all go back to before the crash ever happened, Kate and Jack look at each other and smile. We realize that, although their relationship has always been rocky and neither of them have been perfect, they were always meant to be together in some way or another. Jate is Fate, baby :)

This is why the geniuses in the world have invented Wikipedia, Lostpedia, a whole bunch of LOST fansites and Netflix so you can find out yourself.