My older sister is a Third Order Carmelite. She's also married, a mother of four children, and a pharmacist. Obviously she's not a nun or religious sister, but if not, then what is she?
Many of us learn from friends that they've joined third orders or otherwise describe themselves as tertiaries. Franciscans, Dominicans, Benedictines, Carmelites, and other religious communities extend their identity to lay people in a "third way" that doesn't include clerical status nor communal living in a religious house. The tertiary designation is a secular association that even has official recognition in canon law: "Third Orders - Associations whose members lead an apostolic life and strive for Christian perfection while living in the world and who share the spirit of some religious institute under the higher direction of that same institute are called third orders or some other appropriate name." (CCC #303)
Third orders aren't the only kind of lay associations mentioned in church law. The earlier Code of Canon Law from 1917 recognized lay confraternities (like the Confraternity of Christian Doctrine) and pious unions (think St. Vincent de Paul Society, Knights of Columbus) (see old CIC #700). The distinction between confraternities and pious unions isn't about purpose so much as ecclesial establishment and oversight. Current canon law only formally recognizes tertiary groups while noting the right of other private and public lay associations to form. Needless to say, not every group formed by a Catholic can utilize the name Catholic without competent ecclesial authority. (Heaven knows how the "brand" might be extended otherwise.)
Tertiaries get the nod of approval precisely because of their deep connection to long-established communities with full canonical oversight. The traditional "first orders" were male religious, not necessarily ordained. "Second orders" were composed of women religious. Any layperson who chose to share in the spirit of these communities without taking vows were called oblates by the Benedictines, and third orders by Francis of Assisi.
Today, tertiaries are divided into two categories: secular third orders and regular members. Seculars are like my sister, who lives a relatively normal life while participating in the prayer life and values of her chosen affiliation. She wears a scapular to remind her of her promises, and when she dies, she may choose to be buried in the habit worn by her group. Regular third order members take simple vows as well as following the rule of their community. If a religious community feels resonant for you, inquire about the possibility of associate status.
Books: The Tertiaries Companion - A Prayer Book For the Members of the Third Order Secular of St. Francis of Assisi, by Vincent Schrempp OFM (Franciscan Herald, 2022)
Rule of the Third Order of the Servants of Mary, Servites Third Order (Ulan Press, 2012)