From the Humantities and Social Sciences dept at the University of Edinborough, we get this wonderful description of relations

Transitivity

A relation R is transitive if and only if (henceforth abbreviated "iff"), if x is related by R to y, and y is related by R to z, then x is related by R to z.

For example, "being taller than" is a transitive relation: if John is taller than Bill, and Bill is taller than Fred, then it is a logical consequence that John is taller than Fred.

A relation R is intransitive iff, if x is related by R to y, and y is related by R to z, then x is not related by R to z.

For example, "being next in line to" is an intransitive relation: if John is next in line to Bill, and Bill is next in line to Fred, then it is a logical consequence that John is not next in line to Fred.

A relation R is non-transitive iff it is neither transitive nor intransitive.

For example, "likes" is a non-transitive relation: if John likes Bill, and Bill likes Fred, there is no logical consequence concerning John liking Fred.

Symmetricity

A relation R is symmetric iff, if x is related by R to y, then y is related by R to x.

For example, "being a cousin of" is a symmetric relation: if John is a cousin of Bill, then it is a logical consequence that Bill is a cousin of John.

A relation R is asymmetric iff, if x is related by R to y, then y is not related by R to x.

For example, "being the father of" is an asymmetric relation: if John is the father of Bill, then it is a logical consequence that Bill is not the father of John.

A relation R is non-symmetric iff it is neither symmetric nor asymmetric.

For example, "loves" is a non-symmetric relation: if John loves Mary, then, alas, there is no logical consequence concerning Mary loving John.

Reflexivity

A relation R is reflexive iff, everything bears R to itself.

For example, "being the same height as" is a reflexive relation: everything is the same height as itself.

A relation R is irreflexive iff, nothing bears R to itself.

For example, "being taller than" is an irreflexive relation: nothing is taller than itself.

A relation R is non-reflexive iff it is neither reflexive nor irreflexive.

For example, "loves" is a non-reflexive relation: there is no logical reason to infer that somebody loves themselves or does not love themselves.