Example sentence #1:
"The big, gray, ugly, hungry shark ate the whale."
Who did the eating in this sentence? The shark did.
So, the "shark" is the "simple subject".
The complete subject is "the big, gray, ugly hungry shark".
So the complete subject includes the simple subject, plus all of it's modifiers or descriptive words.
Example sentence #2:
"The cold, white snow fell on the ground".
Simple Subject: "snow"
Complete Subject: "cold, white, snow".