Asthma is a common, chronic inflammatory disorder affecting the airways. There are many types of asthma, and many potential causes and triggers, but common to all cases of asthma is ‘over activation’ of the immune system leading to excessive inflammation. Probably the best resource I’ve found online that covers this and many related questions is:
Look especially at the ‘Health Advice’ and ‘Research’ tabs. Ultimately what determines whether a person will get any form asthma depends on a combination of their genetic profile and the things they are exposed to during life.
Other animals can also suffer from immune-mediated inflammatory airway diseases – these have various different names; examples include feline asthma and equine recurrent airway obstruction.