Time travelling, simply put, is using the "reset clock" feature in the options menu to change the time and date. This is useful for getting things to happen immediately (e.g. getting fruit to reappear after you've picked it all, getting mail sooner, and so on). In order to do this you will need to pick up the phone in your attic before you wake your character up and click 'Reset Clock'.
If you time travel backwards so a character doesn't move, don't try it. It doesn't work.
Weeds come up when time traveling to the future but traveling to the past is easy to get something from Nook's Store that you missed and so on.
There are some negative side effects of time travelling. The most obvious is massive weed growth; if you time travel far into the future you will find your town covered in weeds and the residents will be unhappy. Tom Nook's turnip prices will never go above 96 bells, and Joan will only sell them over 96 bells so you never make a profit from white turnips. Your favourite characters may be more likely to move away.
To fix these problems, synchronise your AC:WW clock with your Nintendo DS clock. They should go away.