When you enter a square for the first time in a day, you have the option to search it. You can do this even if the zone is marked as Depleted, you'll just have less of a chance to find quality items. After you Scavenge the first time, a 2 hour timer will start. After 2 hours, you will search the zone again without having to hit the button or even be online. This is known as autosearching.
If your rucksack is filled while you are autosearching, the next time that you come online, the extra items that did not fit into your rucksack will be dropped onto the ground. You will keep finding things even if your rucksack fills, you just can't carry them with you.
You can only Scavenge/Autosearch a single square once per day. If you are in square 1/0, decide to scavenge, autosearch for a few hours, then move to 2/0, if you move back to 1/0, you will not be able to scavenge 1/0 again. You'll have to wait until the next day.
It's handy to have your rucksack as empty as possible while autosearching, if only because it'll allow you to carry as much as possible back to the town and minimize the amount of AP other people will have to spend coming back to pick up dropped items.
If you become a Hero and choose the Scavenger class, then autosearching will take 30 minutes less and you also have a larger chance to find useful items.