An inverter board is an electronic circuit which boosts a low voltage DC current in to high voltage. It's used to power a CCFL tube in he television, which enables the backlight. The inverter, is composed of a MOSFET transistor, acting as a switch, an a coil, which boosts the voltage. The common high voltage output of the inverter board is around 800V.
If your inverter board has gone bad, you could replace the transistor and the coil in the board. This method is the cheapest option, but involves a lot of soldering and time. I encourage you to replace the whole inverter circuit, which is far easier, and not so much expensive.
You will have to take apart the television. Unplug the TV from the power outlet-wait around 8 hours to give time to the capacitor to discharge. Unscrew all the bolts of the back panel, and then, with a plastic tool-like a credit card- pull the back of the screen. After that, you will be able to see the circuit board of the TV. Localize the inverter board, it should be identical to this:
Then unplug the BUS cable and every CCFL tube connectors carefully-they are easily identified with a pink and a white cable. Afterwards, unscrew the fixing bolts of the board, and change it for the new one. You can buy a new one from ebay:
All you have to do now, is to reverse your steps. Connect each CCFL tube, put the back of the television on, screw the bolts and you are done.