Dogs and cats can either be purebred or they can be mixed breeds. The term "purebred" simply means that the parents are of the same breed, so they conform more closely to what's called a "breed standard". You have a good chance of knowing the general characteristics of a purebred dog or cat: you'll better know what they look like and their general behavioral traits. So is adopting a purebred or mixed breed better?
When you're adopting from a shelter, you will likely see more mixed breeds than purebreds. Mixed breeds are a combination of different breeds. It's possible to make an educated guess about the size, appearance, and temperament of some mixed breed pets and there are even DNA-testing kits available these days.
But like with anything in life, nothing is guaranteed - every dog is an individual (including purebred dogs), and even if you can generalize about traits you'd expect to see with their breed, that doesn't mean it will accurately portray a specific, individual dog.
Some people are absolutely set on getting a purebred pet. The reasons for doing so are varied, but one that crops up often is that purebreds are always healthy and have good temperament. This is a myth. Purebreds are only certified to have come from a certain lineage.
Both purebreds and mixed breed pets can be equally wonderful pets. There is no guarantee in either case whether or not that pet will have health issues down the line. One theory is that mixed breeds have an advantage, being from more than one type of breed and making them less prone to genetic defects commonly found in certain types of purebreds. Ultimately, every dog is an individual regardless of breed. Adopting an adult or a senior pet is a great option - you'll know the pet's temperament, size, and what they look like, things you wouldn't necessarily know when meeting a puppy or kitten.
Researching breeds can be helpful before deciding on one, including learning about common dog breeds and their health problems. Your new pet will also need to match your lifestyle, of course; see choosing a family pet for more information.
If you choose to get a purebred, please consider adopting from your local animal shelter. It's a common misconception that shelters only have mixed breed dogs and cats. This is untrue; shelters quite often have purebreds too. Please read about why adopt from an animal shelter.