The term "transitional fossil" is used at least two different ways on talk.origins, often leading to muddled and stalemated arguments. I call these two meanings the "general lineage" and the "species-to-species transition":
"General lineage":
This is a sequence of similar genera or families, linking an older group to a very different younger group. Each step in the sequence consists of some fossils that represent a certain genus or family, and the whole sequence often covers a span of tens of millions of years. A lineage like this shows obvious morphological intermediates for every major structural change, and the fossils occur roughly (but often not exactly) in the expected order. Usually there are still gaps between each of the groups -- few or none of the speciation events are preserved. Sometimes the individual specimens are not thought to be directly ancestral to the next-youngest fossils (i.e., they may be "cousins" or "uncles" rather than "parents"). However, they are assumed to be closely related to the actual ancestor, since they have intermediate morphology compared to the next-oldest and next-youngest "links". The major point of these general lineages is that animals with intermediate morphology existed at the appropriate times, and thus that the transitions from the proposed ancestors are fully plausible. General lineages are known for almost all modern groups of vertebrates, and make up the bulk of this FAQ.
"Species-to-species transition":
This is a set of numerous individual fossils that show a change between one species and another. It's a very fine-grained sequence documenting the actual speciation event, usually covering less than a million years. These species-to-species transitions are unmistakable when they are found. Throughout successive strata you see the population averages of teeth, feet, vertebrae, etc., changing from what is typical of the first species to what is typical of the next species. Sometimes, these sequences occur only in a limited geographic area (the place where the speciation actually occurred), with analyses from any other area showing an apparently "sudden" change. Other times, though, the transition can be seen over a very wide geological area. Many "species-to-species transitions" are known, mostly for marine invertebrates and recent mammals (both those groups tend to have good fossil records), though they are not as abundant as the general lineages (see below for why this is so). Part 2 lists numerous species-to-species transitions from the mammals.
Transitions to New Higher Taxa
As you'll see throughout this FAQ, both types of transitions often result in a new "higher taxon" (a new genus, family, order, etc.) from a species belonging to a different, older taxon. There is nothing magical about this. The first members of the new group are not bizarre, chimeric animals; they are simply a new, slightly different species, barely different from the parent species. Eventually they give rise to a more different species, which in turn gives rise to a still more different species, and so on, until the descendents are radically different from the original parent stock. For example, the Order Perissodactyla (horses, etc.) and the Order Cetacea (whales) can both be traced back to early Eocene animals that looked only marginally different from each other, and didn't look at all like horses or whales. (They looked rather like small, dumb foxes with raccoon-like feet and simple teeth.) But over the following tens of millions of years, the descendents of those animals became more and more different, and now we call them two different orders.
There are now several known cases of species-to-species transitions that resulted in the first members of new higher taxa. See part 2 for details.