I have an old 6 speed bike (6 cogs on the freewheel in back, 2 chainrings on the front, used to be called 12 speed), and I was able to replace the old chain with a new 8 speed chain. The only problem is that once I did that, the new chain would skip on the old cogs in back due to the mismatch of having a new chain running with the old worn cog teeth. Then I had to replace the old freewheel with a new one (only $15). Since my old chain didn't really have visible corrosion, I probably could've saved the money and just rode it until it started skipping on its own/started to rust/exploded. On the plus side, I now have a really shiny new freewheel and well-lubed new chain...
One tip not mentioned, don't assume that the chain you had on previously was the right length. The one I had was actually one link too long. There are a number of ways people determine the right length, I used 3 outta the 4 listed in Leonard Zinn's book and that's how I found out my old chain was too long.
The Park Tool how-to's are awesome: