When you standardize a variable, you transform it so that it becomes a z-score: its mean becomes 0 and its standard deviation becomes 1. This is useful if you will be combining variables that are on very different scales (e.g., score on the SAT, which has an average of 1000 and a maximum score of 1600, and GPA, which ranges from 0 to 4.0). If you took the simple arithmetic mean of SAT and GPA, your resulting score would be dominated by the SAT score because it is so much larger. To make each variable contribute equally to the mean, you can standardize them.
Creating standard scores in SPSS
In SPSS, you can standardize variables by selecting Analyze -> Descriptives, moving the variables you want to standardize into the "Variables" window, and checking the box at the bottom that says "Save standardized values as variables."
When you press "OK," SPSS creates a new variable for each of the variables you selected. These new variables will generally have the same name as the old variables except they will begin with the letter "z" (because they have been transformed into z-scores).
To create an average out of these standardized variables, make sure you select the new z-score versions of the variables when you do Transform -> Compute.