"Righteousness" is a translation of the Greek word dikaiosune which is the character or quality of being righteous or just. It is ultimately an attribute of God, therefore it speaks of God's absolute perfection. He is righteous in all His ways, and holy in all His works. For the Christian it is moral and spiritual perfection.
First of all we need to be declared righteous by God, since we are all sinners. When a sinner repents and believes on the Lord Jesus Christ, God "declares" that person to be righteous, by giving him/her the "robe of righteousness" which is Christ's. Christ's righteousness is "imputed" to us and we are "justified" by grace through faith (Rom 5:1).
To act righteous, we do what Christ did, speak as Christ spoke, and think as Christ thought. In order to do this, God gives every believer the gift of the Holy Spirit, and it is God who works in you both to will and to do of His good pleasure (Phil 2:13). We are to simply "present our bodies a living sacrifice" to God (Rom 12:1, 2).
Which also means we are not "to be conformed to this world" -- we are not to be influenced by the world's standards and the world's way of thinking. That means we are to be controlled by (1) the written Word of God and (2) the Holy Spirit.
Taking a stand for righteousness means speaking out against sin and evil (a) in the world , (b) in the church and (c) in the home. John the Baptist spoke out against the sins of Herod and lost His life.