I think I can expand on the answer by @boo2060.
The evolution of female mate choice depends on females achieving higher fitness by choosing certain males over others. At the broadest scale, there are two mechanisms by which this can occur, direct benefits, and indirect benefits.
These are material things that (surprise) directly benefit the female. This might be the provision of food by the preferred male, access to a good territory or shelter that is under the control of the preferred male, protection by the preferred male, that sort of thing. Females will accordingly favour male traits that advertise potential direct benefits. These might include male size/condition/colouration, or even built structures like those of the bowerbirds. Basically, these traits show that the male knows how to look after himself, and by extension, he knows how to look after her (as males in good condition generally inhabit the best habitats and will offer the best protection and parenting).
These don't do much for the female, but they increase the genetic quality (and thus potential fitness) of her offspring. This is a much more complicated area of mate choice, and can be split into a bunch of (non-exclusive) hypotheses (below). And of course, attractive males may well offer both direct and indirect benefits.
Sensory bias: Males may, by chance, possess a trait that holds intrinsic appeal for a female. The best explanation might be that it mimics the colour or shape of a food source (e.g. fruit) that the animal eats, and thus the animal has pre-evolved preference for that colour or shape (e.g. red patches). From that point, males with more of that trait, whatever it is, will tend to be more attractive, for no rational reason. This hypothesis is sound, but is currently lacking good empirical evidence.
Fisherian sexy sons: This can in fact lead on from an initial sensory bias. Females may choose males that they find 'sexy', because their offspring will also be sexy, and thus be successfull in reproduction. This can lead to 'runaway' selection for sexier and sexier sons, at considerable survival cost for males. In this case, natural selection and sexual selection work in opposing directions.
Indicator mechanisms: This is the handicap principle or 'good genes' principle. If a structure is costly to produce or maintain, or increases the risk of predation, and yet the male possessing it is still alive, he must have good genes to have withstood such a handicap. It may also advertise that the male is not infested with parasites. The extravagant plumes of birds may be largely explained by this one, but also sexy sons and sensory bias.
Genetic compatibility: Individuals seek partners that are genetically compatible. A certain proportion of the fitness of the offspring depends on the specific combination of genes (i.e. it's not additive), and individuals that can signal or detect signals in such a way that they can choose compatible partners will have an evolutionary advantage. The best example is scent (e.g. preference for dissimilar MHC immune genes - see the human t-shirt experiment).
These mechanisms are well known in theory, but not always well-supported by evidence (yet). This is probably the best reference.