I'm a bit unclear on what you're asking, but it sounds like you've heard about Y-DNA and mtDNA, which are only for male and female lines. As Ole says, the test you took, auDNA, is good for matching all your biological relations.
All such relations who have been tested on AncestryDNA - out to at least 2nd cousins, once removed, or equivalent - will show up somewhere on your match list. The chance of matching a 3rd cousin is about 90%, and that probability decreases rapidly as you get more distantly related. At the same time, the number or relatives you have at those levels usually expands exponentially, so it's believed that most of your matches will be 4th or 5th cousins.
What exactly do you mean by a "DNA cousin" here? Do you mean you have relative who you know were tested on AncestryDNA, and aren't showing up on your match list? Or are you talking about people on your DNA match list?
In a reply to a question below, you report have 1Cs on your mothers side in your matches. If they are within the right range of cM, that gives you DNA confirmation of not only your mom, but also for your grandparents. (If they're NOT in the right range, which is unlikely, you may have stumbled upon their actually being biologically related in an unexpected way - for example, they could be half 1Cs instead of 1C).
I'm not completely clear what you're saying about your father's side in that reply. Are you saying your grandmother was tested (it doesn't really sound like it)? It sounds more like relatives through your grandmother are showing up. If you have strong matches (over 90cM) to 3rd cousins on that side, you can confirm back to your applicable great-great grandparents, by current WikiTree rules. Otherwise you might have to settle for less far back.
As Ole says, further back usually requires triangulation, which is an advanced topic.