Deciding what living things should be classified in the same groups requires deciding what's related to what, and how close those relationships are. Long ago, it was often done by lumping together analogous traits: features that are used to do the same function. This is why, in Biblical times, if they were streamlined and swam ("Beasts of the Water"), or had wings and flew ("Creatures of the Air"), they were classified in the same groups. Certain simple forms were also used for grouping: by this approach, snakes would be grouped with earthworms and eels.
As more and more people studied Nature in detail, it became obvious that a butterfly's wings were very different structures than a bird's wings. And sometimes, it could be seen that two structures used for very different functions - such as a human hand, a bat's wing and a whale's flipper - all contained the same basic internal architecture, with changes in parts producing the outward changes. Traits with similar internal structure are called homologous traits, and it was eventually decided that these traits were a better measure of relatedness than analogous traits. Keep in mind, however, that traits can be both analogous and homologous (like a monkey hand and a human hand), it isn't automatically an either / or situation.
There are, in modern taxonomy, two somewhat different approaches to putting together "family trees" of organisms. In systematics, branches occur when one species splits into two - at the point of a common ancestor to the new branches. Cladistics is similar, but the focus is on when certain "new," special traits arise - humans might split from chimps, for instance, at the point that our ancestors began walking upright. Both are applying fairly rigid "branchings" to points that in real time probably were more spread out, and only sometimes do they seem at odds with each other. Their representations also tend to differ - systematics, the older approach, usually uses fluid, naturalistic branching diagrams, while cladograms tends to branch at geometrical angles and use a lot of straight lines (it was developed when computer printers dealt well with straight but not so much with curvy lines). There is a third approach that might be added in - molecular evolutionary taxonomy is concerned with when certain key genetic differences arise. They tend to be an offshoot of cladistics, but that connection may get less clear over time.
Much basic taxonomy is still done anatomically, although the level of detail has gotten smaller through the use of microscopes and broader through the discoveries of genetics and biochemistry. These will be covered later as they come up in the historical journey.