Utrecht said:
Yes, you are right - actually in a strict Catholic sense, all church goers should wear headgear.
Actually, men are not to wear anything on their heads. St. Paul specifically forbids men covering their heads when praying or prophesying (1 Corinthians 11:4), while commanding women to cover their heads (1 Cor 11:5).
Traditionally women wore hats, chapel veils (called so because in Ireland there were no Catholic churches, only chapels, and much of the history of the Church in America is tied up with Irish immigrants), or mantillas (these are longer than chapel veils and reach the shoulder).
Believe it or not, women were prohibited from entering a church or chapel without a headcovering until 1983, with the release of the new Code of Canon Law. Of course the enforecement of the law on the subject had fallen by the wayside long before. I know my wife (b. 1973) has never worn a head covering in church, although she says it was quite common when she was growing up.
The hat which is worn by the Pope, bishops, and other prelates is called the zucchetto. I am not certain of its theological significance. The Pope wears a white zucchetto, while bishops wear purple and monsignors (priests who have acquired a certain dignity of office) wear black with red piping. Cardinals, of course, wear red -- when the Pope names a cardinal it is often said that he is given the "red hat."
Other "headgear" for Catholic priests and bishops includes the biretta (a four-cornered hat with three ridges and a pom on top -- priests wear black, cardinals red), the miter (worn by bishops and some abbots -- "mitered abbots"), and the Papal crown (a triple crown to signify that the Pope is the highest power on Earth -- kings wore single crowns and emperors double crowns). No Pope has worn the crown (sometimes known as the tiara) since Paul VI laid it on St. Peter's chair at the close of Vatican II.
Personally, I attend Mass in the traditional Latin rite and most of the women at our chapel do wear head coverings of some sort (usually chapel veils, but occassionally a hat or mantilla). It is entirely voluntary, but I personally think it is quite beautiful. Unmarried women generally wear white, while married women wear black or some other color. Widows generally wear black, I believe -- but again, there is no requirement in this regard. My daughter (3 years old) has a pink chapel veil. The most popular shapes of chapel veils seem to be round, triangular, and semi-circular.
EDIT: corrected spelling of "zucchetto"