Your shifter is kaput. Very common problem with the G25 shifter which has never been fixed, and is unlikely to be rectified now thanks to the G27 carriving at sometime in the future (DAMN YOU LOGITECH!).
If your G25 still has a warranty, then give logitech a call and get them to send you a replacement. They will get you to cut the end of the power/usb cable off and send them proof that it is detroyed and unusable, then they will send out a new one.
If it's
not under warranty, then you could try fixing it yourself. Its not like you can make it worse.
Heres what you do. Take off the shifter knob (flip off the silver cap, and undo the screw) and then get an alan key, and remove all of the alan bolts surrounding the gator. This should lift off easily.
Once this gator is off, there will be a white multi connection block at the base of the shifter bar. It will have a bunch of wires going into it. If it's not already visible, get an alan key (or some other small tool) and
very lightly give the wires a tug and see if any have broken from the soldering. Then find out which wires have came from which connection (it should be fairly obvious, as they wouldn't have moved much) and then re-solder them in. Don't put it all back together yet, but plug it in and try it out. If it all works correctly, then put it all back together and enjoy!
Now, theres also another place the shifter could have broken. If it has not worked after repairing the wires at the base of the shifter, then it could have gone in this other place.
Remember the wires i mentioned above that are connected to the multi connection block at the base of the shifter? Well, all of these are then rolled into one, and join upto a connection port on the electric board near the directional pad. It's not entirely common, but the connection block sometimes breaks away entirely from the board, which then takes some of the internal wiring with it. If this has happened, then it is beyond repair, and the shifter is scrap.
To get the shifter apart to check, its fairly simple. Take the clamps off by undoing the small bolts that the clamps thread on (Look at the clamps from underneath the shifter. There easy to spot. There inline with the clamps), then undo the shifter brackets by undoing the screws at the top of it, visible through the shifter gator. Then undo all screws that are on the underside of the shifter. Be careful when taking apart the shifter, as there is a metal ball bearing and spring which will try and shoot out when you take it apart, as will all the buttons at the top. Take it easy and slowly when taking it apart, and remember how the shifter brackets come apart, so that you remember how to put it back together again. There's also alot of grease inside there too, so make sure you have some to put back on (Just incase you accidently take alot off when messing about with it.) and also make sure you have some kitchen roll so that you can cleanup any mess you make.
I hope i've helped. Good luck getting it sorted, whatever route you decide to take!