The Day of the Dead wasn't celebrated in northern Mexico during the 20th century because the people there thought the holiday was mixing pagan elements into the catholic faith.
On the days of the event (November 1st and 2nd), families and friends of the deceased leave food, drinks, pictures, candles, flowers, and possessions of the deceased in "ofrendas" or offerings at their graves.
On my grandma's altar, I would put a drawing that I made because she always liked it when i drew for her. I would also put tamales because she always let me help her make tamales, even though I wasn't very good at it. Lastly, I would put a lavender scented candle on it because she always had one burning.