There are two basic steps to breathing: inhalation and exhalation, or breathing in and breathing out. When inhaling air through the mouth, air will pass more quickly into the lungs. The faster the air passes into the lungs, the more quickly the air can be exhaled. When air is exhaled from the lungs, it passes over the vocal chords, creating sound. So, if you're breathing through the mouth, you can make sounds in a more fluent motion---beneficial to a singer. One other benefit of breathing through the mouth when singing has to do with the positioning of the singer's tongue. Breathing through the mouth causes the tongue to immediately assume a low position, which helps the singer to achieve the desired sounds while singing.
There is only one real disadvantage when it comes to breathing through the mouth while singing. When inhaling using the mouth, it becomes easier for the throat to dry out, which can be detrimental to anyone who sings. Obviously, when the throat is dry, it becomes more difficult to produce sound. This is why many singers rely on throat sprays and teas to treat the throat before and after a performance.
Breathing through the nose is often viewed as being less advantageous for singers than breathing through the mouth, but there are some benefits to choosing nose breathing. Breathing through the nose is considered to be better in terms of hygiene, and it is also believed to do a better job at protecting the lungs, which are vital in the production of sound.
The soft palate is a part of the mouth that creates a block between the nasal and oral cavities. Trained singers are taught that the soft palate should be kept high in order to maximize space and improve sound. It can be difficult to control the soft palate when breathing through the nose, though it's not impossible. One other difficulty to nose-breathing has to do with basic performance. Just like it is difficult to breathe through the nose while speaking, it can be difficult to breathe through the nose while singing.