Many gas appliances now use electronic ignition instead of having a standing pilot light. In other words, they no longer have that small flame that's always going; instead, they have an igniter that ignites the gas. It's essentially an on-demand pilot light. Electronic ignition can break down just like any other mechanical part, but it's not always easy to tell whether it's really in need of repair or something else is interfering with its operation.
Check the Circuit Breaker
An electronic igniter is going to work on an electrical circuit, so check the breaker box for your home. It's possible something tripped the circuit, so while power remains on in the rest of your home, that one circuit leading to the furnace could be off. Try resetting the switch; if it trips again (either immediately or within a few days), call for furnace repair and have the circuit inspected. The reason for the tripping could lie with either the furnace or the wiring.
Check the Furnace Filters
If the breaker seems fine, another simple issue might be the filters. Your furnace has to draw in air to function, and if the air supply is slow or cut off by a clogged filter, that could prevent the furnace from turning on like it's supposed to. Changing the filter could fix this problem; if the current filter looks like it's even getting close to being full, change it and see if that helps. It should be a relatively easy switch, as you can pull out most furnace filters and push the new one right back into the space. If not, call a furnace repair company.
Watch the Igniter
The problem could also be with the ignition module itself. When you turn on the heater (or when it turns on because the thermostat has reached a pre-programmed setting), the igniter may make clicking sounds or start to glow. Watch the igniter as the furnace turns on. If you see and hear nothing, then the problem is likely there. This is not something you should try to switch out yourself unless you have previous furnace repair experience. Let a professional do it.
In any case, if the simple fixes don't work, you need a trained technician to inspect the furnace and figure out where the problem is. Don't try to dismantle the furnace yourself; that's dangerous, especially for gas furnaces. Keep the furnace off and make that call instead.