There are many different types of lightbulbs that contain different chemicals and materials. Depending on the contents of the lightbulb they will need to be disposed of differently. If you are using energy-efficient light bulbs you are able to recycle these. Have a look in your local area for recycling sites that will put these bulbs to good use and stop them from going to landfill.
Incandescent lightbulbs can be thrown out with the regular rubbish. They don't contain any chemicals that will cause an issue within the normal waste disposal system. They also can't be easily recycled due to their construction.
Much like Incandescent lightbulbs, halogen lightbulbs can be discarded in the general waste. They are unable to be recycled easily so the only option is to send them to landfill.
CFL Lightbulbs can't be thrown out in the general waste like Halogen and Incandescent lightbulbs. These Fluorescent lightbulbs contain hazardous chemicals such as mercury and lead. These need to be disposed of properly and handled with care.
Due to EU and US law, these bulbs need to be disposed of via a recycling company. It's also the law that shops that sell these products need to have a recycling area in the store. If you need to dispose of your CFL lightbulb you can simply take it to the store and drop it off in their recycling bin.
If you break one of these lightbulbs in your home, you should shut off all air conditioning and clean up the debris with protection such as a mask and gloves. After doing so take it to your nearest recycling site and dispose of it there. After cleaning up you can safely turn your air conditioning back on.
LED lightbulbs can be thrown out in the general waste as they contain no harmful chemicals. With this in mind, you can also recycle LEDs, as LEDs come as separated bulbs within an overall bulb these still working LEDs can be used again. If you want to do your bit for the environment and save these from going to landfill, search in your local area for LED lightbulb recycling.