The most common way to determine your dog's breed is to simply compare the dog to breed pictures until you find one it resembles. There are also dog breed DNA tests available, but the accuracy of these tests may vary.
Although there is a bit of guesswork involved, most breeds have fairly distinct looks, so mixed or unknown breed dogs often can be identified visually. More experienced people are often more accurate than beginners, because they may notice subtle details. You can try asking your veterinarian's opinion of your dog's breed during an appointment.
Another good resource is breeders and kennel clubs. These may be of more help if you can provide some educated guesses. You can contact local breed clubs or dog show organizations and ask if any of their members are willing to speculate. Dog trainers also see a lot of different breeds and may be able to help.
Canine DNA tests are fun and can be accurate for some dogs. However, some companies have fewer breeds in their databases, so they may be less accurate. Most companies are also only accurate with popular breeds, while rarer breeds may not be included. Mixed-breed dogs with many ancestors may also get inaccurate results because of the large number of influences in their genes.