What are the "echoing footsteps" in Book II, Chapter XXI of A Tale of Two Cities?

In a metaphoric style, Charles Dickens describes the passage of time in Lucie Manette/Darnay's life, who listens to "the echoing footsteps of years." She becomes a mother, loses a male child and hears the "rustling of an Angel's wings" amidst the other echoes that only rarely repeat the "actual tread" of Sydney Carton, a guest of a few times each year, but one whom little Lucie loves.

Among these echoes are "dreadful" ones, rumbling with menace at the time of little Lucie's sixth birthday. An anxious Mr. Lorry, too, speaks of the mania of the Frenchmen to place their money in the English branch of Tellson's Bank.  These footsteps are those of the French peasants in Saint Antoine who have joined with others; they carry pitchforks, bricks and stones stolen from walls. And, at the "center of this whirlpool of boiling waters" is Defarge's wine-shop. It is the beginning of the French Revolution of 1789 and a note written by the prison in the North Tower is saved.

What do the echoing footsteps represent in Chapter 21 of A Tale of Two Cities?

The echoing footsteps that Lucie hears ominously foreshadow the coming of the French Revolution, when hordes of starving, desperate people will take to the streets to demand radical political change. The footsteps represent anonymous, ordinary people going about their everyday business, but who in due course will participate in a momentous historical uprising against the old regime.

At the same time, the footsteps also foreshadow the great violence and bloodshed that the Revolution will unleash. The feet of the revolutionary masses will be clad—for the most part—in shoes, and it's notable that Dr. Manette makes shoes for a living, an activity he started to keep his mind occupied while he was rotting away in the hell-hole of the Bastille. That Manette engages in this occupation under both the old and the new regimes indicates a certain degree of continuity between the ancien regime and its more radical successor. The chaos and disorder of the revolution to come will lead to one tyranny being replaced by another.

It is Lucie who hears the echoes of the footsteps but she has a firm belief that these echoes are positive and they have no negative effect on her.  The way that she views life, even after the death of her son and the incredibly hard times she has faced remains positive and so the echoing footsteps too are positive in her eyes.

Yet they are meant to represent the marching steps of the revolution that has begun in France as the working class has finally thrown caution to the wind and decided to bring down the aristocracy.  The peasants have stormed the Bastille and are killing noble Frenchmen as fast as they possibly can, their blood is running freely.

The noise of the footsteps reaching Lucie is perhaps a symbol of the great power of the revolution and that its effects will be felt not just at the Bastille but also in London and other faraway places as the common people have finally thrown off their shackles and risen to fight.

What do echoes represent in A Tale of Two Cities?

A Tale of Two Cities is largely about influences and interconnections.  The echoes represent the consequences of characters’ choices and influences.

Consider the corner of the house where characters seem to sit and think.  It is a corner for echoes, because the one sitting there can reflect on the events of the past, choices, and the potential of the future.

It is no coincidence that Dr. Manette's house is mentioned as the spot for the echo.

The corner has been mentioned as a wonderful corner for echoes; it had begun to echo so resoundingly to the tread of coming feet, that it seemed as though the very mention of that weary pacing to and fro had set it going. (Book 2, Ch 6, enotes etext p. 64)

Darnay, the character whose past always seems to catch up to him, comments about this.

I have sometimes sat alone here of an evening, listening, until I have made the echoes out to be the echoes of all the footsteps that are coming by-and-by into our lives. (Book 2, Ch 6, enotes etext p. 66)

Darnay is often a victim of others’ choices, but he made the choice to marry despite his past.  He realizes that he can never truly be free of his past, and the echoes will catch up with him.  When Carton responds that their lives will be crowded, he is foreshadowing the trauma and pain that is ahead of them with Darnay is targeted by the revolutionaries.

Consider Lucie’s thoughts on the echoes.

For there was something coming in the echoes, something light, afar off, and scarcely audible yet, that stirred her heart too much. (p. 136)

The events of their lives, including their daughter and the events when her doomed family returns to France, are foreshadowed.

The echoes get more insistent.

But, there were other echoes, from a distance, that rumbled menacingly in the corner all through this space of time. And it was now, about little Lucie’s sixth birthday, that they began to have an awful sound, as of a great storm in France with a dreadful sea rising. (p. 137)

This is the beginning of the end of the family.  All of the choices of the past, including those of Dr. Manette, Charles, and Sydney, are about to catch up to them.