A (reasonably) concrete example may make the point clear. Dolphins, marlins and the sadly extinct marine reptiles called icthyosaurs all have similar profiles, and it's reasonable to think that they do so because that shape is an efficient one for moving a large animal through water fast enough to catch and eat much smaller animals. To test this idea comparatively, we need to look for four sorts of marine animals: those with a streamlined, dolphin-like shape who are large and pursue small, fast fish; those with the shape which do not fill that niche; those which do fill that niche which do not have that shape; and those which neither have that shape nor fill that niche. The naive approach is to total up all the species under each heading we can find, fill out what the statisticians call a "contingency table," and then apply one of the standard statistical tests to see whether or not there is a significant correlation between having a dolphin-like shape and filling a dolphin-like niche. The problem is that those tests all assume that the data-points are independent of each other. In this case, however, related species are not independent samples: all dolphin species resemble each other not just because they occupy similar niches, but because they share a common ancestry. If the ancestral proto-dolphin had both the shape and the niche, then we expect most of its descendants to have both, and so the association between the characters would have to do with descent and not natural selection. In that case, we shouldn't count all the modern dolphins in our contingency table --- but how many dolphins should we count? Just one, for the common ancestor? More than one, to reflect the fact that the traits haven't become detached in subsequent evolution? If so, how many more than one? And what do we do about, e.g., the common ancestor of whales and dolphins?
The problem to solve, then, is to find some way of dividing up our information about contemporary species into statistically independent chunks, and it's clear that doing this right will need information about phylogeny (which species are related to which others, when they diverged, etc.), ancestral characters, and the dynamics of evolution. We need phylogenies to know which species are related, so that we don't count them as independent; we need to know ancestral traits so that we can figure out what has evolved when; and we need to know evolutionary dynamics to get an idea of how often we should expect "chance" (i.e. non-adaptive) associations. (Knowing phylogenies and ancestral traits also lets us test ideas about the direction of evolution --- do marine animals first become predators and then get dolphin-shaped, or vice versa?) It's important to realize that these models of random evolution are just null hypotheses --- which does not mean that we expect most evolution to be non-adaptive!
Statistical methods which take proper account of phylogeny and evolutionary dynamics have only really been developed within the last twenty-odd years. In part this is because of the rising availability of really reliable phylogenies, especially from comparing the DNA and proteins of different species, and using the known rates at which they accumulate random errors --- "molecular clocks." (The number of such publications in molecular phylogeny is currently doubling every two years.) Harvey and Pagel are two of the leading figures in applying the new data and the new models to comparative studies, and this book is their introduction to the newer methods.
They begin with a chapter on why comparative studies are desirable, and why they are hard. This is followed with a chapter which seeks to convince comparativists that they ignore phylogeny at their peril. That done, one wants to know how to construct a phylogeny, and more importantly how to figure out the characters of ancestral species from those of existing ones. Unfortunately, proper phylogenetic methods are the subject of some of the most vehement and technical controversies in modern biology. Harvey and Pagel's advice on this point is basically to leave the controversy to the specialists, but to go with molecular data when you can get it. (They also describe the dangers of trying to use standard taxonomies as substitute phylogenies.)
Chapter four describes the "comparative analysis of discrete data": qualitative characters which are described as taking one of a few different states. (Our initial example, of marine-animal shape and niche, would be an instance of this.) The key idea here is that, while the characters of related species are not independent, the changes in characters along separate lineages are. After a quick survey of previous methods which use this idea, Harvey and Pagel present one of their own, which takes into account the fact that more evolutionary change should be expected over long periods of time than over short ones, and estimates the rates at which traits are changing in the species of interest. Chapter five goes through a similar analysis for continuous traits, advocating the method of "independent comparisons" (using the differences between pairs of species as independent variables), and elaborating null models based on Brownian motion and randomizations of the original data. (We could apply these techniques to our marine example if we made use of some of the quantitative measures of shape which the morphologists have developed.)
Chapter six is on allometry --- the quantitative relations between different parts or traits of the same organism (at one time or at different times), or between the corresponding traits of different organisms. This is an old subject, going back to D'Arcy Thompson and even Galileo, but one which is still far from exhausted. Part of the interest of the subject comes from the idea that allometric relations in themselves explain some evolutionary changes: since metabolic rate goes up in proportion to the three quarters power of body mass, then (so the argument goes) if a lineage evolves to double its average body mass, it will automatically evolve to increase its metabolic rate by a factor of 1.68 and change. Harvey and Pagel are skeptical of this, and rightly so: very, very few allometric relationships can be explained as simply a matter of physics. There is usually some optimality criterion involved in their explanation, though sometimes it's not made explicit. The three-quarters law for metabolism and the many other quarter-power laws, for instance, seem to be consequence of minimizing the energy needed to deliver nutrients and oxygen to the body's tissues by pumping fluids through tubes. "[S]trong allometric trends may often indicate the presence of strongly correlated selective pressures" (p. 177), so the comparative methods developed earlier can be fairly directly applied to refining allometric studies, which our authors proceed to do in some very nice worked examples. (They also provide a good discussion of the different means of estimating allometric parameters, but this does not depend on their comparative methods.)
There's been a lot of work on comparative methods in the last eight years, but this book is still a fine introduction to the modern, statistical-phylogenetic approach. Readers will need some knowledge of both evolutionary theory and statistics (say an intermediate-level course in both), but not expertise. The writing is clear throughout, with technical asides being, in fact, set aside in boxes, and there are even hints of a sense of humor (e.g., quoting the young Richard Lewontin on the near-optimality of all animals). This book will be very useful to anyone seriously studying evolution, and probably to others as well --- it's easy to imagine adapting its methods to historical linguistics and other studies of social change. The only objection to this book is its price, which fails the xerox test by a factor of two.