I have always wanted to join the Navy, as my Daddy had(even though he hated it, and my uncle before him. My Uncle was the Master Chief Petty Officer of the Navy!), but neither were officers. I was lucky enough to have my school paid for because I could play football, but now I am 28 yrs old and have 2 kids. I was told my my local recruiter that I must get a waiver because of my dependents. What needs to be in order for that waiver to be passed? on a second question: what do I need to score on my entrance exam to be eligible? My GPA was not the best as playing football and keeping a job hurt my study time.
If you are talking about an officer program, Brett's answer is right on. If you are talking about an enlisted program a waiver for two dependants is necessary and not a big deal to get. All you do is have an interview with an officer (back in the day anyway, may be delegated to a CPO now).
As to test scores, for officer you take the astb and your scores have to be very good, especially if your gpa is low. On the enlisted side, you take the asvab and the scores required vary depending on the program you want. I expect any college grad will score high enough on the asvab to qualify for a boat load of career options.
Bottoms line. Make sure you are talking to the right guy for what you want. See an officer recruiter for officer programs. I expect the guy you are talking to is an enlisted programs recruiter based on the dependant waiver comment.