Your car will start with or without a alternator in theory. So its hard to say. Does the car turn over or do nothing when you turn the key to start? If it does nothing make sure your battery is all charged and good. Try a jump if in doubt. Also check if it starts in neutral. This could mean your neutral safety switch is going bad or your shift linkage is not engaging in park which is common.
If its turning over but not starting, you have electricity. Now you need to make sure your getting fuel and air. Listen for the fuel pump is a easy way when you turn your car to accessories. Should hear a mechanical noise from your gas tank when key is turned to accessories. (Be quiet and keep the door open) no noise, no fuel. You can test for air by simply removing throttle body forward and hold open your tbody when you go to start. If it starts now, something was impeding the intake. You can check the spark by pulling a plug while the engine is running and holding it against the block. If you see a bolt of electricity between the two, you have spark. If not, no spark. Also do the regular check all the important fuses. Ecu, fuel pump, etc.
EDIT: Also just cause you hear the fuel pump doesn't mean your getting fuel. Its just often the fuel pump is most susceptible so you check that first when in doubt of fuel. You can also pull out a fuel injector and see if its dripping a bit.
