Off the bat, there are at least two types of equality which are obvious and considered desirable by Americans:
Equality in front of the law: Ideally everyone should be treated the same way for the same crimes, regardless of ethnicity, gender, income level, etc...Consider Judged Jean Boyd in Texas, who sentenced Eric Miller, a poor 16 year old black kid, to 20 years of prison for killing a person during a DUI, but Ethan Couch, a rich 16 year old white kid, to 6 months in rehab for killing 4 people during a DUI. This is a blatant case of unequal treatment under the law, and those who complain about inequality in America want to prevent this type of unequal treatment.
Equality of opportunity: Harvard philosopher John Rawls offers the idea of justice as fairness as an answer to the inherent tension between equality and freedom. It is not reasonable to expect that everybody get equal pay or equal recognition. However it is acceptable that everybody receives equal basic liberties and equal opportunity. The principles of Justice as Fairness are:
First Principle: Each person has the same indefeasible claim to a fully adequate scheme of equal basic liberties, which scheme is compatible with the same scheme of liberties for all;
Second Principle: Social and economic inequalities are to satisfy two conditions:They are to be attached to offices and positions open to all under conditions of fair equality of opportunity;
They are to be to the greatest benefit of the least-advantaged members of society (the difference principle). (JF, 42–43)
A key statement is in the second principle "They are to be attached to offices and positions open to all under conditions of fair equality of opportunity;": This means that there is nothing wrong with paying a doctor more that waiter, but everybody has to have a fair shot at becoming a doctor.
Again, those who complain about inequality in the U.S argue that this is not the case in our country: People from lower income and minority backgrounds don't have access to the same educational resources that privileged people do, so the condition that "offices and positions [are] open to all under conditions of fair equality of opportunity" is not satisfied, and we won't have equality until it is.