A laptop that won't respond to its own power button naturally horrifies its owner. Panic won't help you figure out what's wrong with your laptop--thinking about it and taking action will. You may still be able to recover your data and repair your machine without needing to send it in for expensive repairs, but you must develop an action plan.
When a laptop stops turning on, it can mean any number of things. But the most likely culprit is some kind of hardware issue. Check the battery first to see if it is seated properly. If not, remove it from its seating and push it firmly back until you hear an audible click. Check the reference that came with your laptop or the manufacturer's website if you are uncertain of how to do this. If this does not fix the issue, try removing the battery entirely and running the laptop plugged in. If this works, it's likely that the battery is the problem and should be replaced.
If the laptop still doesn't work, try replacing the adapter. Purchase a new one at a discount laptop supply website like Spare Parts Warehouse, linked in the references below. If your laptop's lights do not turn on at all when you plug in your existing power cord, that could be the issue.
Re-seat your laptop's individual components. Shocks and other dislocations can unseat hardware components and prevent your laptop from booting up properly. Even if none of the components appear to be displaced, re-seating may get your laptop to turn on again properly. Consult your laptop's manual for instructions specific to your machine and for diagrams that will help you identify each component.
Smell your laptop. If you detect the distinctly pungent scent of burnt silicon, it's likely that your laptop has become damaged by overheating. Follow your reference manual for instructions on how to open up your laptop casing. Look for melted components. If an isolated component has melted, you may be able to replace it and have your laptop working again without needing to send it in for extensive repairs. If the melting isn't localized, you may have to either have it repaired or purchase a replacement.
If your laptop still doesn't work, recover your data. You may need to either install your laptop hard drive in another laptop or purchase a laptop hard drive adapter kit from a computer supply store. That will enable you to transfer your data off of the laptop hard drive onto another one. To ensure that you do not lose your data in the future, back up your information regularly on a CD or DVD or onto a networked hard drive. Mobile hard drives are more vulnerable to failure than other ones. Transport your laptop in a proper carrying case to protect it from shocks and bumps.