Skeletal muscle refers to any muscle that is not smooth or myocardium (ie most muscles).
In relation to respiration: the internal intercostal muscles cause the ribs to move upwards and outwards (in a 'bucket handle' and 'pump handle' motion), and the external intercostal muscles reverse this process. Neither of these sets of muscles are generally needed during quiet breathing. The diaphragm is the most used muscle in respiration, during quiet breathing this contracts to flatten and increase the thoracic volume, hence pulling air in through the reduced pressure. This is inspiration. Expiration involves the relaxation of the diaphram and the abdominal contents pushing upwards to reduce thoracic volume. Also used more rarely are the accessory muscles, eg sternocleidomastoid and serratus anterior which increases the expansion of the ribcage, or the various abdominal muscles which expell air forcibly.