Where in the "ranks" of taxonomy (domain, kingdom, phylum etc.) do common words used to describe animals such as amphibian and reptile fall?
This is all a matter of how we define terms, and which definitions are scientifically useful.
Traditionally (since Linnaean times), organisms were classified based on their outward similarities (phenetic systematics). As such, the traditional definition of reptile, which is the same as any common usage that I've heard, includes organisms like snakes, lizards, turtles and crocodiles but does not include birds. This is the class Reptilia. As evolutionary understanding increased, however, it was found that some reptiles, like crocodiles, are actually more closely related to birds than to other reptiles. This makes the common use of reptile a paraphyletic classification.
In modern phylogenetic systematics, organisms are classified in monophyletic groups, or clades, that include an ancestor and all of its decedents (ie it is based on evolutionary relationships). Thus the class Reptilia, since it is paraphyletic, is essentially meaningless in this system. In order to have consistent nomenclature, you could either say that birds are reptiles (a new, cladistic meaning of the term, but which contradicts traditional/common usage) or come up with a new name that includes all traditional reptiles and birds. Taxonomists taking the latter approach use the term Sauropsida for this clade. Sauropsida does not have a taxonomic rank like class because such ranks are useless in cladistics (and actually quite arbitrary even in the traditional system).
So, to specifically address your question:
In phenetics, both Reptilia and Amphibia have the taxonomic rank of class.
In phylogenetics, what we commonly consider reptiles and birds are grouped in a single clade that can be called Sauropsida or Reptilia (re-defined to include the clade Aves) depending on who you ask or which paper you read. Amphibia form a clade.
Perhaps the most important take-away message is that taxonomy is largely un-standardized.
Amphibians form a class in phylum Chordata.
Reptiles, when considered as a taxonomic group, are also usually seen as a class in phylum Chordata. However, some scientist don't see reptiles as a valid taxonomic group because they are not monophyletical and modern taxonomy tend to accept only monophyletical groups.
That is, according to the taxonomy I was taught at school some decades ago, fishes, amphibians, reptiles, birds and mammals where classes in phylum Chordata. Nowadays, it is known (and stressed) that some reptiles are evolutionary closer to birds than to some other reptiles: there were common ancestors of crocodiles and birds that weren't ancestors of turtles.
Since monophyletic groups are defined as all descendants of a common ancestor, reptiles defined in the usual sense aren't a monophyletic group because they don't include birds. Therefore, some taxonomist either expand reptiles to include birds (and then "birds" and "reptiles" can't be both classes, although remaining valid) or just discard reptiles as a valid taxonomic group.
In the other hand, extant amphibians seem to descend from a common ancestor, and therefore they aren't questioned as a valid class.
Interestingly, fishes, regarded as a class for long time, are also being questioned as a valid taxonomic group, since some fishes share a common ancestor with tetrapodes (terrestrial vertebrates) which is not an ancestor of some other fishes.