<p>I have sets of values in rows in Excel
First row contains weights of how much each value in each column contributes to weighted total.
Now, I want to mark in each following row the value that contributed to the weighted total the most, by changing its background. I played with conditional formatting, but my Excel skills are modest, to say the least.
So, how do I do that?</p>

Here's what I think you are trying to do: You have a list of numbers (they happen to be weights, but that's not important). You want the largest number to be highlighted differently.
Suppose the cells are in the range A1:A8.
Select the range A1:A8.
Select Conditional Formatting (Format | Conditional Formatting)
Enter the condition Cell Value is/Equal To/=MAX(A1:A8)
Click the Format... button and set the Pattern to affect the background.
(Tested on Excel 2003 on Windows.)