Border Terriers were developed near the English-Scottish border to travel with foxhounds and go to ground after foxes. That job called for a dog small enough to work in tight places, tough enough to cover uneven country, and willing to keep going. Modern Borders may spend more time on the sofa than in a field, but many still have the curiosity, speed, and prey drive that made the breed useful.
At home, a Border is often affectionate, alert, and ready to join in. It may enjoy a game, a sniffy walk, a training session, and then a nap close to its people. This is usually a more people-focused terrier than a dog that wants to be left in the yard to make its own fun. An under-exercised Border may find its own project, such as digging, barking at a window, or checking every weak spot in a fence.
Temperament is never guaranteed by a breed name. Meet the individual dog and ask clear questions about its comfort around visitors, children, dogs, cats, handling, noise, and time alone. A terrier that is easy with one familiar cat may still chase a squirrel or an unfamiliar small animal outside.