Your hygienist will do units of scaling...scrub off the tartar with special instruments.
Like it was said, she/he will do it on all your teeth. You can see it more on the front teeth because it's easier for you to see it there but if you haven't had a cleaning in a long time, you probably have tartar on your other teeth as well.
It should take about 45 minutes but yeah, its really important for you to get a cleaning.
For the cost, it depends how many units of scaling, polishing is done...and whether you will have a fluoride treatment.
When I had 3 units of scaling, one half unit of polishing, and a fluoride treatment, it costed me 160$ CAN (I have no insurance)
I went for another cleaning this week and I payed 115$ CAN for 2 units of scaling and 1 unit of polishing.
The scaling is what is the most expensive...its about 45$ per unit...something like that. But it has to be done