1. grammatical auxiliaries (be, do ,have) are part of grammatical constructions, but carry little meaning. (be followed by an -ing participle marks the progressive aspect, be followed by a past participle marks the passive voice, and have followed by a past participle marks the perfective aspect.)
2. modal auxiliaries (may/might, can/could, shall/should, will/would, must, ought to) are not part of grammatical constructions, but express modal meanings.
lexical verb (or main verb): a verb which refers to an action, activity, event, or state, and is capable of being the main verb in a verb phrase.
finite verb: a verb which is marked for tense (present or past) or modality. A finite verb phrase is a verb phrase with a finite verb in it. There can only be one finite verb in a verb phrase, and unless the verb phrase is simple, the finite is always the (first) auxiliary. All modals are finite. A verb in the imperative is also finite.
non-finite verb: a verb which is not marked for tense or modality. The non-finite verb forms are the infinitive, the past participle and the ing-participle. E.g. (to) write, written, writing. Non-finite forms can combine with each other in non-finite verb phrases, e.g. having written, having been written, being written. Non-finite forms can also combine with finite ones in finite verb phrases (in which case the finite verb comes first), e.g. has been writing, (he) had been writing, (the book) will have been written.