If I go from a 195/55/15 to a 205/50/15 will that in any way effect my odometer and speedometer?

 

using a tire calclator, at 60mph your spedo would be reading 60.97 with the 205

So there isn't any dramatic changes to your spedo or odometer.

http://www.dakota-truck.net/TIRECALC/tirecalc.html

 

The outer diameter of the 205/50-15 is 1.6 percent smaller, so your odometer and speedometer will both read 1.6 percent higher than with the 195/55. Whether that's important or not is up to you. There are usually other, more significant reasons for choosing one of these tire sizes over another aside from the tire size itself - differences in cost, availability, etc.