What does it mean if when I breathe it feels like I have a cough in my lungs?

Inflammation. It means you likely have inflammation of some sort in your airway. Why you have it is not clear from the question - asthma, reflux, bronchitis, pneumonia, exposure to chemicals, irritating dust..... See your physician and discuss with him/her.
Chest congestion. When you feel like you want to cough, it usually means you have some secretions in your bronchi and you need to clear them. This could happen from a viral bronchitis type infection, probably the commonest cause, or, if you aren't sick and it happens frequently after certain exposures, you could have allergic asthma. Cough is very often the main symptom of asthma, not wheezing. Get checked soon.