What kind of steroids are inside an inhaler?

Corticosteroids. The type of steroids in inhalers used to control asthma are corticosteroids (cortisone-like). These are not the kind of steroids that are sometimes abused by athletes (called androgenic steroids), so you don't have to be concerned about having those side effects. Still, we must respect corticosteroids; if not used as prescribed, they can cause their own side effects.
Depends on type. There are two basic types of inhalers - relievers and controllers. Relievers generally contain bronchodilator medications which help to get immediate relief of cough, wheeze or breathlessness. Controllers usually contain a corticosteroid to help get the inflammation that causes asthma symptoms "controlled" which will decrease the amount of wheeze, cough and breathlessness over time.
Glucocorticoid... The class of steroids in inhalers are the anti-inflammatory glucocorticoid steroids.