We can trace the Temperament Of The Lhasa Apso back to its original purposes -- it breeding, and to natural evolutionary factors.

The early Tibetans were hunters and herders and they could depend on the characteristics of the native Tibetan dogs to hunt, guard flocks, and to serve as companions and guards inside their tents. The Dalai Lamas originally breed Lhasa Apsos in the sacred city of Lhasa (Tibet's traditional capital and home of the Dalai Lamas) be sentinels (i.e., watchdogs): they guarded the monasteries and warned the large Tibetan Mastiffs (a typically ferocious and aggressive breed, at least in Tibet) of approaching strangers. Together, the Lhasa Apso (with its acute hearing and natural suspicion of strangers) and the Tibetan Mastiff (with it's willingness to confront predators the size of wolves, leopards, and humans) made for a formidable combination against attackers.

Its sentinel characteristics tend to make the Lhasa Apso alert, intelligent, and independent. Its sense of loyalty and its strong watchful and protective natures make it excellent companions for a family. The Lhasa Apso can appear "aloof" at times, but this can be attributed to its watchful, attentive and protective nature. On the downside, because of its watchdog nature, it possesses a sharp, loud and persistent bark, and it is naturally suspicious of strangers, compounding its potential barking problem. However, the Lhasa Apso will respond to good motivational training if it has a strong dominate owner.
Because the Lhasa Apso evolved in a cold climate, to conserve heat, it will not waste energy with any nervous or hyper-active behavior. Rather it tends to be calm and sedate. Because of its independent sentinel nature, the Apso is not an overly affectionate breed. Rather it is watchful and protective nature makes it less likely than other breeds to engage in play with humans or other dogs.

The Apso really just enjoys being in the same room or area with its family, and it tends not to tolerate or interact with rough, badly behaved or over active children.
Most owners and breeder report that the Apso has a very gentle temperament and are almost distinguished in their demeanor. This is probably why the Buddest monks and Dalai Lamas considered them the perfect companions. Because of their long hair and gentle temperament, the Lhasa Apso enjoys being petted and groomed, and they tend to prefer to be in the company of humans rather than other dogs or other animals. While they tend to bond strongly with one individual, they do like to be with lots of people -- once they are familiar with them of course.
Overall, the Lhasa Apso's qualities combine to make a great companion dog for a single person or a family.