Auxiliary means to help or support something else, to act as a backup, and to be secondary to something more significant. Auxiliary verbs are often called helping verbs.
Lexical means "relating to words." Lexical verbs are the main verbs in a phrase or sentence. To put it simply, lexical verbs are any verbs which are not auxiliary verbs. Lexical verbs carry their own meaning, (full verbs), and therefore can use but don't necessarily need a helping (auxiliary) verb.
- I was acting the jerk yesterday at the carnival.
"Was" is the auxiliary verb and "acting" is the lexical verb because it describes the state or action in the sentence.
- If we go to that restaurant, I will want the lobster.
"Will" is the auxiliary and "want" is the lexical verb.
Some examples of auxiliary verbs are: be, is, was, may, have, had, can, could, would, did, might.
Some examples of lexical verbs, which are much more numerous, are: run, think, see, walk, go, pull, make.