The characters literally combine to mean “Whole Family - Joyful Happy.” It's a direct and powerful expression of a wish for complete family happiness.
“阖家欢乐” is deeply rooted in the core Chinese value of family-centric collectivism. Unlike in many Western cultures where individualism is often emphasized, traditional Chinese culture places the family unit at the center of society. The happiness, reputation, and well-being of the group are paramount. This phrase is most prominent during festivals that champion family reunion (团圆, tuányuán), such as the Spring Festival (Chinese New Year) and the Mid-Autumn Festival. During these times, family members travel vast distances to be together. The wish of “阖家欢乐” is therefore not just a pleasantry; it's a blessing that affirms the most important social value: a harmonious and complete family.
This is a formal and somewhat traditional phrase. You would not use it in casual, everyday conversation.