With, perhaps, one or two exceptions, most members of the dog family are instantly recognisable as such. Ignoring the domestic breeds, they typically have muscular bodies, long legs, large, mobile, ears, heads that are broadly triangular in shape, and bushy tails.
Their legs are the shape they are because they're adapted to chasing prey, and most dogs are therefore pretty good at running. They have four toes on each of the hind feet, and, while most of them do have the full set of five on the front feet, the thumb (or "dewclaw") doesn't reach the ground. Their snout is the length and shape it is, partly to get in a large and sensitive nose, and partly to fit in an array of teeth that don't restrict them purely to eating meat.
Teeth are particularly important when we come to classify mammals, especially when all you've got to go on is an incomplete fossil. So it's worth taking a brief look at their arrangement in the dog family. There isn't, in fact, one universal pattern here, and they do vary a little between species, but, in general, dogs have 42 teeth, with 10 on each half of the upper jaw, and 11 on the lower. At the front are six clipping incisors in each jaw, and, just behind them, the large flesh-tearing canine teeth that take their very name from the animals.
Behind this, they have a full set of premolars, all of them sharp and somewhat jagged in outline. The last premolar in the upper jaw is, as is typical of carnivorans, adapted into a great flesh-shearing carnassial tooth, slicing through meat like a butcher's knife, and working against the carnassial in the lower jaw, which is formed from the first molar. The first molar in the upper jaw is also fairly large, typically with three stabbing points on it, but, while they certainly fare better than in the hyper-carnivorous cats, the remaining ones are fairly small, with the third molar in the lower jaw being tiny, if it's even there at all.
The dog family has been remarkably successful with, thanks to the existence of the dingo, wild representatives on every continent except Antarctica - apart from mice and bats, there are no other land-dwelling mammal families that can say that. There are a number of reasons for this great success, which tend to boil down to dogs being highly adaptable animals.
Those teeth are one sign of that; compared with other carnivorans, there are quite a lot of them, and they aren't highly specialised. This means that dogs can eat pretty much whatever they want to - even if they clearly prefer fresh meat - and gives evolution a broad template on which to build, so that, when we do find more specialised species, they aren't all specialised in the same way. Other advantages include the superb senses of hearing and smell that dogs have, and the ability of many of them to cooperate in hunting to degrees that few other carnivores can match.
The remaining anatomical peculiarity of the dog family is likely familiar to anyone who has bred the domestic form. Most mammals (apart from higher primates) have a baculum, a bone inside the penis, but it's particularly large in dogs. It's there to aid penetration, but the unique thing isn't that, but the presence of a bulb of erectile tissue near the base of the penis that expands during copulation to lock, or 'tie', the animals in position for the duration of the act. Once in place, it's really rather difficult to get it out again until everything's over.
The dog family used to be divided into two sub-groups, which we can roughly describe as "wolf-like" and "fox-like". As our understanding of genetics has improved, we have been able to piece together a more detailed picture of how the living species evolved, and it turns out that this apparently obvious distinction is rather over-simplified. Now, we consider the dogs to be divided into at least three groups, and possibly more.
Because, despite the physical resemblances that lead to almost all of them looking "dog-like", the family is a diverse one in terms of habitat and behaviour. They live in almost every habitat, from the frozen north to the baking deserts, and from grassy plains to rugged mountains. Despite their reliance on chasing prey, there are even some in tropical jungles. Generally speaking, they are all sociable animals, but that manifests in different ways, from the highly coordinated pack hunting of wolves to the much smaller family groups of red foxes.
In all, there are 34 generally recognised non-fossil species in the dog family. Six of them are formally listed as endangered, three of them critically so. For that matter, one has already gone extinct.
The formal name for the dog family is Canidae. As with other scientific names for families of animal, this comes from the name of the genus that, at the time of naming, was considered the most typical of the wider group. In this case, of course, that genus is Canis, to which the domestic dog, its wild ancestors, and its closest relatives all belong.
So that's where I'll start next time.
[Photo by "Editor at Large", from Wikimedia Commons. Cladogram adapted from Bardeleben et al 2005 and Lindlblad-Toh et al 2005.]