Cat and /æ/ is pronounced here:
Pronunciation is governed by very complex rules, so the sounds in and around a word can vary greatly with factors we often don't think about: Position in a sentence, whether we are expressing some kind of emotion, if we are placing stress on a word or part of a word for some reason (e.g., "What kind of a cat are you?" could see the vowel sound in cat stretched out in duration and pronounced more loudly.
Secondly, the main difference between native speakers of a dialect and others is in the way our minds impose order on and interpret sounds. So listening carefully and trying to open one's mind to hear more or different kinds of information can be very helpful in improving pronunciation.
There is no substitute for that than having a well-trained or naturally helpful native speaker to listen to you and help you listen to the target sound, and try to describe how and where differences are heard.
In addition to the dictionary samples linked to above, then, I offer a recording of my own voice pronouncing cat in some different ways and at different speeds. Note that "exaggerating" sounds and slowing down sounds can be a terrible way to try to help people hear (perceive) a sound in the way native speakers do, because, well, those just are not the sounds! However, such exaggerations may sometimes help draw your attention to certain features of the "normally" pronounced sounds. Be sure to use those as your guide. And when you get the chance, listen together with a patient native speaker!
(slowly, with some exaggeration)
cat / a a a / cat
I have a cat / I have a cat / cats and dogs / cats and dogs / my cat wears a hat
a / apple / apple / red apple / (exaggerated link d_app : red apple / green apple / do you like red apples or green apples?
candy apple / cat hat
cats and dogs / my cat had kittens.