Clean the Contacts yourself
Usually you can get excellent results from just cleaning the battery contacts. There are a variety of different ways people do this. The general concept is you need some sort of material to scrape/sand/wipe off the crud... and something thin and long to attach it to. My preferred method is to find an ink PEN with an ink ERASER. Ink erasers are usually much more abbrasive then the standard rubber erasers on pencils. Stick the eraser end of the ink pen into your battery chamber, and scrub it accross the battery contacts. With a little work and patience, you should be able to remove the corrosion. Then, just gently shake out any eraser shavings. Other methods include: Using a que-tip dipped in distilled alcohol, using a small piece of fine grain sand paper taped to a pencil, and scraping with the wooden end of long wooden matches. By trying one or more of these methods, you might be able to save yourself a hefty repair bill!
Replace the Battery Cell holder or Battery Cover
Some flashes (and even some cameras) have battery cell holders. These are removable plastic cartridges that typically hold four AA or AAA batteries, and form them into one large power source. Other flashes and cameras have removable battery covers that have metal battery contacts on them. If you can't effectively clean either the battery cell holders or the battery covers, you can usually just replace them!
Take (or Send) the Camera / Flash to a Repair Shop
If neither of the above options work for your equipment, you will probably need to take your camera to a Repair shop to get fixed. I would expect that the average repair would cost between $75-125. In some cases, with expensive equipment, this can be well worth the cost. Othertimes, if you have less expensive equipment... you might want to think about replacing it instead. If you do need to replace your equipment, you can either buy the same exact model (and keep your old equipment for spare parts!) or you can use the opportunity to replace it with a better model.