Keep in mind that when discussing this topic there is no "universal" truth. There are plenty of exceptions to the generalizations, but there are also some overall themes to how we perceive families to be now and how we saw them back in the day. Some of it is perception and some of it is real.
The biggest change between how we view families today and how they were viewed in the 50's and 60's is in our very definition of family. Back then, the ideal image was a family clearly headed by a man who was married to a woman. The man worked while the woman stayed at home. He made the decisions about money and had the final word on most things, while the woman was expected to be the "homemaker." People who were married tended to stay married, even if life together was unpleasant.
Now we have more single parent households. Remarried parents are much more common (think of the Brady Bunch...both parents were remarried, but only because their former spouses had died.) Interracial marriages were frowned upon (sometimes to the point of disowning children!) There were no options for "civil unions."
The family of the 50's and 60's lived in a suburb of cookie-cutter small houses (a generalization, but that's what people back then often thought of as ideal.) There was a more clear distinction between urban, suburban, and rural life. Now, cities often blend together by connected suburban sprawl.
People were expected to marry young in the 50's and 60's, generally shortly after high school, and the idea of having no children (or even stopping after one) would have been very odd.
The biggest change in family structures today probably is the represented by the changed roles for women. More families have working mothers and fathers, and "stay at home dads" are more common (and you'd never have seen that in the 60's!)
Those are, in general, the main family differences: an increase in single parents, an increased role for women in the family, and a broadened definition of what actually constitutes a family in the first place.
While the first answer is not at all wrong, we should not lose sight of one of the biggest changes in today's families. This is the fact that (in some ways due to the factors the first answer mentions) there is a great deal more divorce today than there was 50 years ago. This has led to many more single-parent families or split and/or "blended" families. This trend is much more prevalent among those with less education. More educated and well-off people generally live in fairly traditional two-parent families, but those lower down the economic ladder do not.
It is also very important to note that many more children are being born outside of marriage. This, too, leads to more single-parent families. This is due in part to a greater incidence of couples who cohabit rather than marrying.
The traditional family structure has become much less prevalent today than in the 1960s. The first answer mentions some ways in which this is so. This answer offers some other important changes.
In the early 1960s, the wife and husband occupied traditional gender roles as other and father, respectively. The father was the breadwinner and the mother was responsible for maintaining the household and rearing the children. Beginning in the late 1960s, there was a movement towards womens’ liberation, and marriage rates decreased in the late 1960s and then into the 1970s. However, in today's society, these expectations of womens’ behavior have deteriorated, as more women are joining the workforces. Women are spending more time outside the home and pursuing careers, in addition to raising families.