Public colleges and universities are operated and funded by state or municipal governments. Private institutions are generally either independent or church-related. One type is not inherently better than another, but some distinctions can be made.
State universities almost always charge a fraction of the tuition of private schools for people who reside within the state. At some state schools there is only a moderate tuition difference for out-of-state students, whereas others charge the equivalent of private school tuition for out-of-state students. It is probably accurate to say that state universities tend to have larger class sizes than private schools, but there are exceptions for both types of schools. Generally, a listing of schools that have the best reputations would be dominated by private schools, but a small number of state schools have also achieved the distinction of ranking among the best ten or twenty in the country.