A cross-bred dog is the offspring of two purebreds of different breeds. Mixed-breed (also called random-bred or, less flatteringly, a mutt or mongrel) usually refers to a dog with more than two breeds in its background.
Purebred dogs, cross-bred dogs, and mixed-breed dogs all make excellent companions. If you're looking for a dog for a specific purpose—say hunting, sheep-herding, or top-level obedience competition—then you probably want to choose a breed that typically excels in that area. It may be, too, that a cross of specific breeds will give you the traits you want. For instance, a Labrador Retriever X Golden Retriever cross will likely be a good-natured dog that likes water and loves to retrieve.
Don't be duped into thinking that mixing breeds will result in healthier offspring. The truth is that puppies, whether purebred or mixed, inherit genes from each parent. If the parents pass along the genes for a hereditary disease, the pups stand a good chance of having that disease. Hybrid vigor—the idea that unrelated parents produce healthier offspring—works only if natural selection is in operation. In the wild, a dog with clinical symptoms of a debilitating disease such as hip dysplasia or epilepsy would not be able to survive, and so would not pass on its genes. But in the modern world, with reasonable veterinary care and a constant food source, dogs with serious, even deadly, problems do survive long enough to produce lots of pups.
If you or someone in your family is limited in some way by health issues, you may need to consider that when choosing a breed. Perhaps allergies or a respiratory condition mean you need a dog that sheds very little or not at all. It would be hard to predict how much a mixed-breed pup might shed when it grows up, but some breeds are known to shed very little and, in a few cases, not at all. Other limitations, such as lack of strength to manage a big, energetic dog, could also make it more important to have some ability to predict what a puppy will grow into.
I don't suggest choosing a dog strictly by its looks. That would be as silly as choosing a spouse just because of blue eyes! You live with the whole package, including energy, temperament, and interests, not just looks. But I do think we should enjoy looking at our companions. If you really like long, flowing hair, use that as a trait you desire, and then research the other characteristics of the many breeds with lovely long coats. There may well be one with all the traits that will make a good choice for you. A mixed-breed adult may also fit the bill, but mixed-breed puppies often grow up to look a bit different from what their baby faces lead us to expect!
Temperament and behavioral traits are also much more predictable with purebreds. This doesn't mean that all dogs of a certain breed act the same. As with members of a human family, there can be variations in individual personalities, energy levels, behaviors, and looks within a single litter, let alone the whole breed. But still, responsibly bred purebred dogs do tend to have certain basic traits that are highly predictable. You might prefer an outgoing dog that likes everyone. Or perhaps you prefer a dog that will bark to alert you when someone is around. If a busy dog that's always on the move would drive you bonkers, then you need to select for calmness. But if you want a dog that's always ready to join you in active outdoor pursuits, you need to find one with more get up and go.
Again, many mixed-breed dogs will fit either bill. Just remember that it's harder to predict how a mixed-breed puppy will act as an adult unless you know for sure the breeds in its background. If specific behavioral traits are important to you, then find a properly bred purebred pup of an appropriate breed and from appropriate lines within the breed.
You may have heard that purebred dogs are “inbred” and therefore plagued with inherited health problems as well as flaky, high-strung dispositions. To some extent, this is true of poorly bred purebreds. The reality is that all living things are prone to inherited problems, and dogs are no exception. Purebred dogs do inherit problems, but so do mixed-breeds. There are advantages to a responsibly bred—let me repeat that, because it's really important—responsibly bred purebred. First, you can look for a breeder who tries to breed away from problems known in the breed. Failing that, you can make an informed decision. If you know that a breed is prone to early death from cancer, you can decide whether that is a risk you can live with and whether the short life expectancy is outweighed by the pleasure that a dog of that breed will give you.