It sounds like your battery isn't charged/charging.
Try charging it while the laptop is turned off. If it still doesn't charge, then either the battery is dead and needs replacing or the power cord is the problem.
You can also try pulling the battery and then see if you can run the laptop on the power cord only. This is not recommended to do all the time, but is a good way to check to see if you can get it to turn on. This won't tell you if the cord and/or battery are toast, it is just one step.
The cord could actually be damaged so that it still will power the device, but not charge the battery at the same time.