The very simplistic answer is that dogs and wolves belong to the same genus and are so closely related that they are often considered to be parts of the same species. The genus is Canis (Latin for dog) and the species are lupus (Latin for wolf) with domestic dogs being the subspecies familiaris (...the familiar one) to make the dog's name Canis lupus familiaris and the wolf's Canis lupus lupus. Being this closely related, dogs and wolves can interbreed to produce a hybrid wolfdog, which can themselves be fertile. Fertile interbreeds is one of the characteristics that are used to determine the separation of species. This means that things like mules, which are a cross between a horse (Equus ferus) and donkey (Equus africanus), and are different species within a genus aren't fertile.
Wikipedia has a good illustration of the relationships between dogs and wolves on the page on the genus Canis. Basically the genomes of dogs and wolves will be very very similar, much more similar than that of humans compared to chimpanzees. The last common ancestor (LCA) of dogs and wolves is about 10,000 years ago, at the domestication of dogs.
On the other hand Humans; Homo sapiens (Latin for "man" (as in mankind) and "wise-one") and chimpanzees; Pan troglodites (Pan was a god of the forest in mythology and Troglodites were mythical cave dwellers), are not even in the same genus, though we are in the same family grouping, known as the Hominidae, meaning that we are closely related in terms of body shape.
We don't actually have fossil evidence of the LCA for chimps and humans, so we don't know exactly what it looked like. However, for some frame of reference, I refer you to the excellent and informative A Short History of Nearly Everything by Bill Bryson, which has a line on hominid remains which goes something like this:
If you took all the ancient hominid remains you could comfortably fit them into the back of a pickup truck with room to spare.
While our genomes are similar, the genetic similarity is about 98.8% - which in a genome of about 3.2 billion bases, this equates to about 35 million differences. The similarity between wolves and dogs is 99.9%, which, with a genome of 2.4 billion bases, gives about 2.4 million differences, far fewer than between humans and chimpanzees.
The LCA of humans and chimpanzees is about 10-13 million years ago, which is about 1000x longer than that for dogs and wolves. If you could go forward about another 10 million years, dogs and wolves might be similarly separated to humans and chimpanzees.
Another problem here is the concept of relatedness - the LCA of humans and chimpanzees isn't a chimpanzee or a human, rather it is a species with similarities to both. On the other hand, we know the LCA of dogs and wolves, and know that it was a wolf, because of their close relationship. Thus we can say that dogs are the wolf's little sibling or are friendly wolves, but we can't say that humans are brainy chimps (or that chimps are less intelligent humans)