Diagnosis:
In order to determine if you have asthma, a visit to a healthcare provider (such as a doctor or a nurse practitioner) is the best way. While the internet is a wealth of information, one should never diagnose themselves based on things that haven't been confirmed by a healthcare professional. There are several ways a health care provider can diagnose asthma:
1) Health History
They will ask about your concerns, such as when you noticed you have trouble breathing, what it feels like, when your breathing becomes worse or better, and how long your symptoms have been going on, in addition to other important questions. A hallmark of asthma is when someone wheezes during exercise, when they are sick, are around animals with hair or fur, when the weather changes, or when they inhale dust or pollen, to name a few.
2) Physical Exam
Your diagnosis is also based on an assessment of your lungs, chest, nose, throat, and skin. An x-ray might be done, too. This helps the doctor rule out the possibility of other diseases. Wheezing during regular breathing is a sign of asthma.
3) Breathing tests
You will most likely be asked to do a few breathing tests. These are nothing to worry about. For example, the doctor might give you something called a "spirometer" (a hand-held device that measures how much air you can breathe in), and ask you to take as deep of a breath as you can. They also might test you with a "peak flow meter," which is a small device that you hold and blow into. When you blow out the air, a little line moves and tells you how well your lungs are working. The higher the number, the better (keep in mind that different ages will have different target numbers). You will do this then take medicine that is made to expand the airways. Then you will do the peak flow meter test again. If you are able to blow the line further, [specifically, if the volume of air you can blow out in 1 second increases by 10 or more (National Heart, Lung and Blood Institute & National Asthma Education and Prevention Program; p 65-66)] this means that the medication worked, and could mean you have asthma.
Left: Spirometer; RIght: Peak Flow Meter
4) There might also be allergy testing. Since asthma and allergies are closely related, the doctor might do this to see what could cause flare-ups.
Overall, what a healthcare provider will do is see if a) your symptoms match those of asthma b) your airway constriction is reversible with medications meant to help and c) rule out other causes (like COPD).
Disease Progression:
Diagnosis is important because then asthma control can be planned. Even when a person with asthma does not feel like they are struggling to breathe, there still might be airway inflammation. Over time, this damages the lungs. It can reduce lung function by permanently narrowing the airways and cause airway scarring over time. This is termed "airway remodeling," and occurs over time in an asthmatic, especially if their asthma is not controlled with medication.
Sources:
I found all of my information from the the sources listed below. The U.S. Department of Health and Human Services article is very long, in depth, and research-oriented, but if you like reading things like this is has a ton of information on what asthma is, how exactly it is diagnosed, and how it can be managed.
Asthma and Allergy Foundation of America. (2011) Asthma Overview: Diagnosis. Retrieved January 31, 2015, from http://www.aafa.org/display.cfm?id=8&cont=7
National Heart, Lung and Blood Institute & National Asthma Education and Prevention Program. (2007, August 28). Expert Panel Report 3: Guidelines for the Diagnosis and Management of Asthma. Retrieved January 31, 2015, from http://www.nhlbi.nih.gov/files/docs/guidelines/asthgdln.pdf
The Asthma Center: Remodeling of the Airways. Retrieved January 31, 2015, from http://www.theasthmacenter.org/index.php/disease_information/asthma/what_is_asthma/definition_of_asthma/remodeling_of_the_airways_/