The Binomial Nomenclature (Genus, Species) introduced by Swedish Naturalist Carolus Linnaeus (Karl von Linné), the father of modern taxonomy, in the mid-1700's lasted for 200 years. Linnean classification organizes species in a hierarchical scheme based largely on similarities in their forms and other traits that usually, but not always, reflect evolutionary relationships.

However, the Linnean System was done before Darwin and does not reflect evolutionary relationships. Cladism, a system based on the idea that members of a group share a common evolutionary history, and are "closely related," was proposed by German taxonomist, Willy Henning in 1950. Systematics & Cladistics became popular in the mid-1900-'s.
Since the early 2000's there has been a movement to have a new PhyloCode based on clades replace Linnean taxonomy. Cladistics is now accepted as the best method available for phylogenetic analysis, for it provides an explicit and testable hypothesis of organismal relationships.

See other Clades at the evolution page.

Phylogeny

The connections between all groups of organisms as understood by ancestor/descendant relationships. Uses cladograms, which are like genealogies of species, to express relationships among groups of organisms
See Phylogeny and phylogenetic systematics at Univ. of California Museum of Paleontology (UCMP)
and phylogenetic Trees

Paleontologists

Are interested in understanding life through time is not just at one time in the past or present.

Systematics

- The way that biologists reconstruct the pattern of events that have led to the distribution and diversity of life. Includes taxonomy and Phylogenetics.

Cladistics

a relatively recent conceptual change in taxonomy, which can be best described as an explicit means of expressing Darwinian concepts of phylogeny, descent with modification. A clade is a monophyletic lineage, one derived from a common ancestor, that can be defined by one or more novel shared characters. The resulting analyses produce branching diagrams, cladograms, that visually show phylogenetic relationships, a sequence of common ancestries, among the study organisms.

It is based on genealogical relationship and the branching of the phylogenic tree, but does not consider evolutionary change after the branching. For example the crocodilians branch off the tree closeer to birds than to other reptiles, so the cladist groups crocodillians with birds. Source: www.bio.ilstu.edu/armstrong/syllabi/222book/chapt1.htm (Ill. St. Univ.)
See Also: John Kimball's Taxonomy page.
An Introduction to Cladistics at UCMP

A phylogenetic tree is a specific type of cladogram where the branch lengths are proportional to the predicted or hypothetical evolutionary time between organisms or sequences.

Cladograms cannot be considered completely true and accurate descriptions of the evolutionary history of organisms, because in any cladogram there are a number of possible evolutionary pathways that could produce the pattern of relatedness illustrated in the cladogram. See: Phylogenetic Trees at cnx.org

The origin of the eukaryotic cell was a milestone in the evolution of life. Although eukaryotes use the same genetic code and metabolic processes as prokaryotes, eukaryotes' higher levels of organizational complexity has permitted the development of truly multicellular organisms.

The roots of the eukaryotic cell are still being defined. According to the Tree of Life Web project, 1997 . "There are two alternative views on the relationship of the major lineages (omitting viruses) shown below:

The "archaea tree": 

   ,=============== Eubacteria (Bacteria) [Domain]
   |
   |                    ,== Euryarchaeota [Kingdom]
===|  ,=Archaea [Domain]=|
   `==|                 `== Crenarchaeota-Eocytes [Kingdom]
      |
      `============ Eukaryotes [Domain] (Plants, Animals, Fungi and Protists [Protozoa, most algae] )

The "eocyte tree": 

     ,======== Eubacteria
     |
     |  ,===== Euryarchaeota
=====|  |
     `==|  ,== Crenarchaeota-Eocytes
        `==|
           `== Eukaryotes

See Also: Introduction to Studying Eukaryotic Origins at UCLA

"Evolutionary Classification" :
Superfamily Hominoidea (humans and the apes)
.....Family Hylobatidae
..........Genus Hylobates (gibbon)
.....Family Pongidae ("higher apes")
..........Genus Pongo (orangutan)
..........Genus Gorilla (gorilla)
..........Genus Pan (chimpanzees-two species; common chimp and pygmy chimp (= bonobo)
.....Family Hominidae
..........Genus Homo (humans) 

"Cladistic Classification:"
Superfamily Hominoidea
.....Family Hylobatidae
...............Genus Hylobates (gibbon)
.....Family Hominidae
..........Subfamily Ponginae
...............Genus Pongo (orangutan)
..........Subfamily Homininae
...............Genus Gorilla (gorilla)
...............Genus Homo (two species of chimp and humans)

See Tree of Life Diagrams at the evolution page.