The cheapest and most popular places to eat in Singapore are hawker centres, essentially former pushcart vendors directed into giant complexes by government fiat. Prices are low ($2-5 for most dishes), hygiene standards are high (every stall is required to prominently display a health certificate grading it from A to D) and the food can be excellent - if you see a queue, join it! Ambience tends to be a little lacking though and there is no air-conditioning either, but a visit to a hawker centre is a must when in Singapore. However, be leery of overzealous pushers-cum-salesmen, especially at the Satay Club in Lau Pa Sat and Newton Food Centre at Newton Circus: the tastiest stalls don't need high-pressure tactics to find customers. Touting for business is illegal, and occasionally a reminder of this can result in people backing off a bit.
To order, first chope (reserve) a table by parking a friend by the table, note the table's number, then place your order at your stall of choice. They'll deliver to your table, and you pay when you get the food. Note that some stalls (particularly very popular ones) have signs stating "self-service", meaning that you're expected to get your food yourself - but if it's quiet or you're sitting nearby they'll usually deliver anyway. At almost every stall you can also opt to take away (called "packet" or ta pao (??) in Cantonese), in which case they'll pack up your order in a plastic box/bag and even throw in disposable utensils. Once finished, just get up and go, as tables are cleared by hired cleaners.
Every district in Singapore has its own hawker centres and prices decrease as you move out into the boonies. For tourists, centrally located Newton Circus (Newton MRT), Gluttons Bay and Lau Pa Sat (near the River), are the most popular options - but this does not make them the cheapest or the tastiest, and the demanding gourmand would do well to head to Chinatown or the heartlands instead. Many of the best food stalls are located in residential districts away from the tourist trail and do not advertise in the media, so the best way to find them is to ask locals for their recommendations. And if you miss western food, Botak Jones in several hawker centers offer reasonably authentic and generously sized American-restaurant style meals at hawker prices.