A grade is a taxon united by a level of morphological or physiological complexity. The term was coined by British biologist Julian Huxley, to contrast with clade, a strictly phylogenetic unit.
An evolutionary grade is a group of species united by morphological or physiological traits, that has given rise to another group that has major differences from the ancestral condition, and is thus not considered part of the ancestral group, while still having enough similarities that we can group them under the same clade. The ancestral group will not be phylogenetically complete (i.e. will not form a clade), so will represent a paraphyletic taxon.
In order to fully understand evolutionary grades, one must first get a better understanding of Phylogenetics, defined as "In biology, is the study of the evolutionary history and relationships among individuals or groups of organisms (e.g. species, or populations). These relationships are discovered through phylogenetic inference methods that evaluate observed heritable traits, such as DNA sequences or morphology under a model of evolution of these traits"
The most commonly cited example is that of reptiles. In the early 19th century, the French naturalist Latreille was the first to divide tetrapods into the four familiar classes of amphibians, reptiles, birds, and mammals. In this system, reptiles are characterized by traits such as laying membranous or shelled eggs, having skin covered in scales or scutes, and having a 'cold-blooded' metabolism. However, the ancestors of mammals and birds also had these traits and so birds and mammals can be said to "have evolved from reptiles", making the reptiles, when defined by these traits, a grade rather than a clade. In microbiology, taxa that are thus seen as excluded from their evolutionary grade parent group are called taxa in disguise.
Paraphyletic taxa will often, but not always, represent evolutionary grades. In some cases paraphyletic taxa are united simply by not being part of any other groups, and give rise to so-called wastebasket taxa which may even be polyphyletic.