Source : Webster's Revised Unabridged Dictionary (1913)

Degree \De*gree"\, n. [F. degr['e], OF. degret, fr. LL.
   degradare. See {Degrade}.]
   1. A step, stair, or staircase. [Obs.]

            By ladders, or else by degree.        --Rom. of R.

   2. One of a series of progressive steps upward or downward,
      in quality, rank, acquirement, and the like; a stage in
      progression; grade; gradation; as, degrees of vice and
      virtue; to advance by slow degrees; degree of comparison.

   3. The point or step of progression to which a person has
      arrived; rank or station in life; position. ``A dame of
      high degree.'' --Dryden. ``A knight is your degree.''
      --Shak. ``Lord or lady of high degree.'' --Lowell.

   4. Measure of advancement; quality; extent; as, tastes differ
      in kind as well as in degree.

            The degree of excellence which proclaims genius, is
            different in different times and different places.
                                                  --Sir. J.
                                                  Reynolds.

   5. Grade or rank to which scholars are admitted by a college
      or university, in recognition of their attainments; as,
      the degree of bachelor of arts, master, doctor, etc.

   Note: In the United States diplomas are usually given as the
         evidence of a degree conferred. In the humanities the
         first degree is that of bachelor of arts (B. A. or A.
         B.); the second that of master of arts (M. A. or A.
         M.). The degree of bachelor (of arts, science,
         divinity, law, etc.) is conferred upon those who
         complete a prescribed course of undergraduate study.
         The first degree in medicine is that of doctor of
         medicine (M. D.). The degrees of master and doctor are
         sometimes conferred, in course, upon those who have
         completed certain prescribed postgraduate studies, as
         doctor of philosophy (Ph. D.); but more frequently the
         degree of doctor is conferred as a complimentary
         recognition of eminent services in science or letters,
         or for public services or distinction (as doctor of
         laws (LL. D.) or doctor of divinity (D. D.), when they
         are called honorary degrees.

Source : WordNet®

degree
     n 1: a position on a scale of intensity or amount or quality; "a
          moderate degree of intelligence"; "a high level of care
          is required"; "it is all a matter of degree" [syn: {grade},
           {level}]
     2: a specific identifiable position in a continuum or series or
        especially in a process; "a remarkable degree of
        frankness"; "at what stage are the social sciences?" [syn:
         {level}, {stage}, {point}]
     3: an award conferred by a college or university signifying
        that the recipient has satisfactorily completed a course
        of study; "he earned his degree at Princeton summa cum
        laude" [syn: {academic degree}]
     4: a unit of temperature on a specified scale; "the game was
        played in spite of the 40-degree temperature"
     5: a measure for arcs and angles; "there are 360 degrees in a
        circle" [syn: {arcdegree}]
     6: the highest power of a term or variable
     7: the seriousness of something (e.g., a burn or crime);
        "murder in the second degree"; "a second degree burn"

Source : Free On-Line Dictionary of Computing

degree
     
        The degree (or valency) of a node in a graph is the number of
        edges joined to it.