Beauty: Attractive because of its form.
It is difficult to define "beauty", but the above short formula has the merit of isolating two aspects of what is said to be beautiful that are usually agreed on: What is beautiful is pleasing, interesting, attractive, appealing etc. and this is due to some of the relations between its parts, which are covered by the term form.
There are many philosophical and psychological problems relating to beauty, apart from defining it in such a way that it at least does justice to many of its uses.
Three important ones are: What is the use of it? And on what criterions and underlying processes are such judgments that so-and-so is beautiful based? And is it ever true or false in an objective sense that so-and-so is beautiful?
The first question is especially motivated by the fact that there are all manner of things people think beautiful: Other people, nature, architecture, music, pieces of technology, theorems of mathematics and more. Indeed, it is very difficult to name any man-made item that is in common human use that is not somehow estheticized ("stylized") i.e. given some form that is meant to be beautiful, or at least pleasing.
The question what is the use of experiencing all manner of things as beautiful is also important in the sense that very much of human behaviour is motivated by the desire for beautiful things or being beautiful oneself; that humans go to great, often irrational, lengths to try to achieve beauty of various kinds; and that the feeling that so-and-so is beautiful (or ugly) may be quite strong and compelling, including the giving of a far from humane treatment to humans who are experienced as ugly.
The second question is mostly unanswered, though there are some guesses that relate to harmonies of a mathematical kind, such as the Golden Section, which was already known to the Greeks, or to counterpoint in music, that accentuates a harmony by briefly breaking it.
Besides, it is also clear that something like perceived beauty helps the sexes to be attracted to each other and multiply, which is why it may have arisen in nature, and it is also clear that here, at least, hormones also play a role, indeed to that extent that heterosexual women find attractive in males what idem males does not interest in other males, and likewise for males about women. (Though as far as facial beauty is concerned both sexes agree that symmetry and regularity are important.)
Some part of the question of what causes the judgment that so-and-so is beautiful have a clear answer, cynical though it may seem: Fashion. Indeed, it seems as if people sincerely start to feel that such-and-such - "The New Look From Paris!" - "really is" beautiful when sufficiently many of their mates and the authorities they believe in have confidently asserted it is. (See: Roles)
The third question, that concerns the truth-status of esthetical judgments is interesting for at least two reasons. First, much of what inspires and motivates people - love, marriage, art, music - depends on judgments that so-and-so is beautiful and worthwile. Second, such judgments have the forms of statements - "This Picasso is truly beautiful!", "You must be out of your mind: It's ugly as hell!" - and thus seem to be capable of being true or false.
The Romans already pronounced that "De gustibus non disputandum", but given that esthetical judgments and experiences are quite important for people, and there is for everyone but one world to live in, this seems unreasonable, even if it often is unclear what criterions one uses in one's esthetical judgments, and quite clear that the esthetical judgements of many are based on little else than current fashion.
Even so, it seems that in architecture, in Europe for example, people have for many ages widely agreed on the beauty of certain kinds of architecture - which only ceased, at least ostensibly, in the 20th Century, apparently mostly by propaganda in the media, since when large expanses of utterly boring grey concrete housing have been the norm in cities, at least for those who do not belong to the top income brackets, that usually still like to live in what was also counted as beautiful 200 and 500 years ago.