There are lots of supermarkets and food shops throughout Brighton, and most will have a cheap food range and a posh food range (for fruit and veg pretty much all will have normal and organic ranges). Generally in order of increasing price it goes something like Aldi, Asda, Somerfield, Tesco, Co-op, Sainsburys, Marks&Spencers and Waitrose, though trying to order them is pretty contentious. There's also a local supermarket chain called Taj Stores which has a couple of branches and seems to have a good range of organic/whole/ethical foods.
In addition Brighton has several health and organic food shops, the biggest of which being Infinity Foods in the middle of the north Laine area (on North Street). Infnity has a very good stock of local and organic food, and also has a very nice instore backery.
There is also Bills further down North Street (more known for local veg) and Sunny Foods up at Beaconsfield Villas - there will be plenty more round and about though.
If you class chocolate as a staple then it's worth mentioning Montezuma's in the South Laine area - they are a local company who do some of the best chocolate in town (their "almost dark" milk chocolate is really good).
The best bread I've found has been in the Real Patiserie (there's at least two branches, one near Brighton Station and one in Hove), though the bread can sell out quite quickly. There are also quite a few chain bakers in town (Forfars, Greggs, Grints) but the bread isn't quite as nice.
There are some good organic/free range butchers in Brighton too, one on Elm Grove called Archers and another on Church Road called Canhams
Again there are many places to eat in town - the one thing I know for sure is that there will be a few gems that I'll have missed from this list. Also I've not put any of the many Thai restaurants down, as the food can be really variable.
Also it's worth mentioning that pub food in the UK has come a long way in the past 20 years, and now there are lots of pubs doing good food (and the pubs are smoke-free now too).
Anyway, here's the current list of place I know:
Pub food
Fish and Chips
There's lots of fish and chip shops in Brighton
Japanese
Chinese
Curry
Cafes
Pizza/Pasta
Fast Food
English
Kebabs