You asked a specific question (what is genetic distance), which is good as it is that which is most relevent to your very specific goal (finding your cousin's father).
Firstly the results from a Y test bear no resemblance whatsoever to Ancestry like autosomal results. You do get a list of matches, and their surnames, who are contactable, but they are listed with their "genetic distance" from the tester. At a VERY basic level, genetic distance is the number of mutations between the two parties Y chromosone. And as mutations only occur from one generation to another, genetic distances are an "indication" of how many generations difference there is between the two testers. A genetic distance of zero is no mutations suggesting the very closest of relationships. Then there is 1, then 2, then 3 etc.
Zero gd is instantly very interesting and you need to contact that person to discuss. A gd of 1 is marginal but it is still worthy of contact. A gd of 2 is pretty useless tbh and this is my closest match. (ftDNA actually suggests in my case we are descended from a common ancestor born in 1600 - 1700 which is absolutely no use to you, nor me for that matter).
Now, the other big difference is that bigger does not mean better. If you test at 37 markers, like me, and get matches no closer than a gd of 2, then testing any more markers up to Y700 is a waste of time in what you are trying to achieve. If I look at my 12 marker results I have 10s of zero gd matches and I have many at 25 markers too. But when sampled over 37 matches I have no zero gd matches. Doing what you are trying to do, then it would be pretty much a waste of money and time me upgrading to more markers in this case.
Now, if you find matches at zero (or 1) gd then it is worth upgrading but ONLY if the other person has upgraded too. This is quite easy as you do not need to retest you just pay an upgrade fee and voila, fresh more detailed analysis.
For what you need I suggest you do a 37 (or maybe a 67) marker test and see what you get. Any zero gd matches and you are coooking on gas straight off the bat. If after chatting to this match, you may want to upgrade, but only after ensuring that they have already done more markers than you or at least intend to.
Like I said before, the distribution of match surnames might be interesting and useful, maybe not.
In addition, you will also get a bunch of numbers and letters that constitute a haplo group. Not relevant in finding your cousins father but to some its interesting.
Good luck