What are the characteristics of the comedy of manners implied in The Second Mrs. Tanqueray by Arthur Wing Pinero? Please give me some hints.

This is a difficult question for two reasons. The first is that Arthur Wing Pinero's play The Second Mrs Tanqueray is a tragic drama, it is not a comedy of manners. The second is that if it has characteristics of a comedy of manners, they resemble the comedy of manners as first innovated by Jane Austen in the earlier Regency era. Pinero wrote in the latter part of the Victorian era. The Second Mrs Tanqueray was published in 1893; Queen Victoria was crowned in 1837.

By this period in the Victorian era, comedy of manners had changed considerably from Jane Austen's innovation of the genre with Pride and Prejudice, Sense and Sensibility and her other four novels. By the 1890s the comedy of manners was best epitomized by Oscar Wilde's The Importance of Being Earnest and The Ideal Husband. The comedy of manners, a then popular form of play, according to Maureen Moran in Victorian Literature and Culture, shared the characteristics of witticism, craftsmanship; bourgeois characters, loyal servants, obstacles; unexpected, clever resolution; frivolity, a formulaic treatment of love and marriage, rapier wit; idioms; contemporary settings and costumes.

To step back to the Regency era and Jane Austen's original comedies of manners, the characteristics of those are a concentration on the character's personality; conflicts between the heroine and society; witty, elegant, intellectual dialogue; a witty, ironic narrator (often omniscient third person but sometimes limited third person) who is amused and sympathetic; complications in the pursuit of love, usually brought about by a character flaw in either one or both the hero and heroine; subplots of love pursuits of realistic but in some way ridiculous characters; realistic adverse social situations (e.g., elopements with officers, raiding Gypsies); well constructed and skillfully told stories; and manners, that is details of the way of life and social customs of that era.  

Since The Second Mrs Tanqueray isn't really a comedy of manners, it seems more likely that you will succeed in finding Austenian-style characteristics of the comedy of manners genre than Victorian-style characteristics seeing as how it doesn't approach the level of one of Wilde's plays.

However, be that as it may, some hints at what to look for come in "The First Act," during the conversation between the Aubrey and the dinner guest once Cayley Drummle walks in. The dialogue shows wit and repartee as they talk about "fish and cutlet" and "pancake" and bandy him for his explanation for missing dinner. Some situational irony develops in Drummel's speech about the newly wed Miss Hervey, now the new Lady Orreyed, plus it is the beginning of the set-up of the realistic social situation that the play addresses with the intention of disquieting Victorian prejudices and convictions.

[For more information on Victorian tragedy and comedy of manners, see Maureen Moran's Victorian Literature and Culture.]