A sacrifice that’s still alive. Obviously.
The least complicated way I can explain a “living sacrifice,” is a person whose actions aren’t self-serving; someone who would deny themselves pleasures or behaviors that would be unpleasing or deplorable to God.
A living sacrifice is an individual who chooses God’s decrees (found in scripture or led by a matured conscience) over their own desires.
An example of this would be chastity before marriage. When a young Christian is extremely physically attracted to another individual, but restrains themselves from having sex (before making a marriage covenant) they are living a sacrificial life.