The packages of paper money and 'gold' we burn. Each package has a name and their relation to the (living) person burning it to them - like an envelope. The incinerator is postal system.
There's food in every festival, celebratory or not. Take today, Ching Ming, which is when we (Chinese people) are supposed to go sweep our ancestor's graves and pay them our respects. We buy loads of paper money, Mercs, DVD machines, mansions, even paper servants, pack them into named paper bags (see above) and 'send' them off by burning them at the grave/temple.
We also bring typical "bai sun" (拜神, literally bowing at the gods) accoutrements - incense, oranges, wine, hard boiled eggs, roast pork (燒肉, yes the kind with the crispy skin)... each family 'standard' bai sun pack differs, according to personal taste, including that of the ancestor who they're visiting. My grandpa apparently has a taste for Shaoxing wine, and/or cognac, so we bring that along too.
But dead people can't really injest anything, so what do we do? Well we eat it for them of course. So this whole 'paying our respects' thing is actually an excuse for a picnic (graves tend to be a little further away from the city). I can imagine people coming to sweep my grave with truffle paste, prosciutto, figs, crispy pork (old trads die hard, and it is good...) and maybe a drop of d'Yquem if they really love(d) me.
Gotta go down some cognac now, so laters.