The evolutionary relationships connecting taxonomic groups are dealt with in the field of cladistics. Ideally extant species are positioned on a phylogenetic tree, or evolutionary tree, according to both morphological and molecular data. Obviously within the fossil record, genetic material is restricted to a few 100,000s years, and it is frequently only the skeletal elements which survive (though occasionally longer in the case of proteins).
In view of this, palaeontological science largely relies on comparing specific morphological traits. Such features are grouped into either plesiomorphies (ancestral, or primitive, traits shared between groups), synapomorphies (derived characteristics shared by related offspring), or homoplasies (superficially similar, but genetically unrelated traits, usually due to convergent evolution). Once indexed, this information is used to construct a phylogenetic tree, in which separate branches form monophyletic groups, which stem from a single ancestor. Nowadays this process is assisted by computational phylogenetics. Sometimes, where very little evidence is available, these relationships are somewhat tentative, and groupings may be subtly altered on the basis of later findings.
In the context of your example, the placental mammals (the Placentalia, themselves a subclass of the Eutheria) and the marsupial mammals (the Marsupialia) are both evolutionary descendants of the Theria mammal subclass. As such, they all share ancestral traits of tribosphenic molars, calcaneums with enlarged sustentacular processes, talus bones with distinctive necks, and several other memorable identifiers. However a subgroup, such as the Placentalia, may be differentiated according to derived features, notably the absence of epipubic bones and pouch, narrow stylar shelves on the upper molars, absence of a shell membrane… and so on and so forth. The systematic application of this technique has given us the present outline of taxonomic relationships where molecular evidence is unavailable.
A basic mammalian phylogenetic tree can be seen here for context.