Unilateral exercises emphasize using a single arm or single leg in the movement. Training unilaterally helps to work on strengthening each side of the body individually and which helps to correct muscle imbalances. It's common for most people to have a dominant side, but it becomes an issue when it starts to overcompensate for the weaker side. The weaker side may try to recruit other muscles instead of the correct muscles for the movement or exercise and this can result in injury. When performing bilateral exercises (using both sides together) like a squat, the stronger leg will carry most of the weight and the weaker side will be underutilized.
The core is a major factor in unilateral exercises because deeper stabilizing muscles have to maintain balance and stability in standing exercises. The core also is recruited during upper body unilateral exercises to prevent the torso from rotating. Single sided training is very effective in rehabilitation because it actually indirectly stimulates the opposite side of the body as well. This is called cross-education of the muscles and it happens in the neural pathways of the brain. Indirectly stimulating the non-working side of the body helps to improve strength in the injured area. The brain stimulates the muscles on the other side of the body to work and this helps to increase strength and improve recovery outcomes.
Check out this link from Women’s Health for some unilateral exercise examples: