Cats are the only animals to choose to allow themselves to be domesticated. Though just how domesticated seems to be a little of a questionable thing..heh.
All cats are different. If this one jumped in your lap he/she has accepted you as their human and will do so until you don't allow them to anymore of it dies. Just give it some attention when it seems to want it...And it will let you know.
Feed it decent food. The cheapo crap doesn't have must to it.
If it's coming inside then it will need a litter box or you will hate the little surprises left for you in really odd places.
They will work a lot to keep their claws sharp. It pays to have something for them to scratch on. Their claws work like those razor knife blades that break off to get to the new parts. They claw on a surface to get the older worn claw out of the way to allow them to use the new sharper one under it.
Most cats are pretty low maintenance but when they want attention they will work hard to get it. Take care of them and they are great animals to have around.