A lexeme is an abstract unit of morphological analysis that gives rise to a set of forms that a single word can take. For example, run, runs, ran and running are forms of the same lexeme RUN.
In HPSG, the type lexeme is introduced in order to be able to express the relationship between the different forms of a word. Since there are many regularities that hold over classes of lexemes - e.g. common noun, proper noun, intransitive verb, transitive verb etc. - the type lexeme is organized into subtypes, on which the generalizations can be expressed in form of constraints. The type hierarchy that classifies lexemes in English can be seen below:
from Sag/Wasow/Bender 2003