You don't feel much like talking now that your fancy new phone is at the bottom of the hot tub, ocean, or--shudder--toilet. But if you retrieve your handset quickly, you might be able to get it working again--if you take action right away.
Remove the battery immediately. If the phone is still on, don't bother turning it off first (or saying "goodbye;" whoever you were talking to has probably hung up anyway). If the battery doesn't pop out easily--I'm looking at you, iPhone--at least power it off as soon as possible. Pop out the SIM card, too, if it has one.
After quickly removing the power source, you can spend a little longer stripping the phone down further. If it's an iPhone, open it by wedging a guitar pick in the seam around its sides. (For more details, visit iFixit, or search on "take apart iPhone"; you should find videos such as those from DirectFix.) On most other phones, remove the plastic shell with an eyeglass screwdriver. Look beneath stickers if the screws are hidden from view.
Wash out the phone, especially if it met chlorinated water or saltwater, which can conduct electricity and corrode parts more easily than tap water. Use electronic circuit cleaner fluid to wash the inside. A bottle from an electronics store costs about $8. (An online source is CriticalCleaning.com.) Alternatively, use rubbing alcohol.
Let the phone air-dry for a couple of hours on a sunny windowsill or other warm area. Don't use a hair dryer--excessive heat can cause more damage. Then submerge the phone in a bowl or plastic bag full of uncooked rice; seal the bowl airtight with plastic wrap (a bag should be resealable). The rice will absorb moisture and help the evaporation process; just keep it out of direct sunlight to avoid condensation. Leave the phone sealed up for at least a couple of days. If you use a SIM card, you can pop that card into an unlocked handset to make calls while you're waiting. (Read how to unlock a cell phone.)
Remove the phone from the bowl or bag and, using a can of compressed air, gently spray off any rice dust. (Push the button in a series of quick blasts, instead of holding it down.) Reassemble the phone, add the SIM if needed, and make a call.
If the phone won't come back to life, you're probably out of luck. But the SIM card (if your phone has one) should still have your contacts stored on it. Try inserting it in a new phone, or bring the old hardware and SIM to your carrier for help; its technicians may be able to recover the phone's address book.