How does Coleridge use allegory in "The Rime of the Ancient Mariner"?

In The Rime of the Ancient Mariner, Coleridge presents an allegorical tale to promote a respect for nature—a popular theme with the Romantic poets in the face of Industrialization in England at that time. 

An allegory is defined as...

...an extended metaphor in which a person, abstract idea...It usually involves moral or spiritual concepts which are more significant than the actual narrative.

It is a tale that shares a life-lesson, which is reflected in the theme of the literary piece.

The Romantic writers expressed themselves through poetry. There were several characteristics common to Romantic writing: a renewed respect for nature; interest in the supernatural; an expression of melancholy; the idealization of women and children; the pursuit of personal freedom, and an interest in the past (particularly the Middle Ages).

Respect for nature is the prevalent theme in The Rime of the Ancient Mariner. To understand what drove Coleridge to write this allegorical tale, look to the historical context of the time. The face of England was drastically changing with advancements made with the Industrial Revolution. Machines had been invented that made cloth and thread more quickly; dresses, for example, were manufactured and sold "off of the rack" rather that being individually made. Mining produced coal—a source of fuel. It replaced the need for wood burning, but destroyed the land. Manufacturing (besides exploiting workers) also poisoned the environment. 

The factories themselves polluted both air and water, belching out smoke from coal-fired furnaces and releasing dye and other wastes into rivers. 

In Coleridge's epic poem, he tells of a man (and ship's crew) who shows his disrespect for nature. The tale describes the fate of ship and crew, but the story's intent was to address mankind's destruction of the natural world through a continued disrespect for nature. 

...it is not his killing of the bird that is wrong, but the mariner's—and by extension humankind's—callous and destructive relationship with nature that is in error.

The mariner is speaking to a Wedding Guest: his ship has set sail. An albatross appears from the mist. The crew considers this to be a sign of good luck—a good "omen." 

At length did cross an Albatross:

Thorough the fog it came;

As if it had been a Christian soul,

We hailed it in God's name. (62-65) 

The Wedding Guest notes a sudden look of despair on the face of the mariner: the sailor admits to killing the bird. In much the same way that factory and mill owners were destroying the English countryside without any sense of responsibility, the mariner killed the bird with his bow—for no apparent reason.

At first the crewmembers censure the mariner for his actions, but when the fog disappears, they condone with what he has done—making them as guilty as the mariner. When the supernatural punishes the crew, only the mariner remains alive—forced to tell his tale—to encourage others to respect nature. To the Wedding Guest he notes:

He prayeth well, who loveth well

Both man and bird and beast.

He prayeth best, who loveth best

All things both great and small;

For the dear God who loveth us

He made and loveth all. (609-614)

Coleridge does not warn against killing birds or sailing, but of harming nature and man being punished. In poisoning the environment, nature cannot nurture mankind. The poet's allegorical tale is a warning of potential disaster if nature is not protected.

The relatively simple seeming plot of the poem, the sailor who destroys a beautiful piece of nature without thinking and in so doing awakes the destructive and vengeful power of the natural world is an allegory for a number of things including some aspects of Christianity as well as a simple philosophy of respect for the natural world, something that Coleridge felt very strongly about.  The beautiful imagery that fills the poem serves to highlight the great beauty and power of the natural world and the horror that follows the sailor's poor decision serves to illustrate the danger of not respecting nature and its beauty and power.

The process of the poem and the way that the death first of the albatross and then the entire ships crew and the destruction of the ship itself then lead to the rebirth and new life of the sailor also mimic the natural process of death and regeneration.

The use of clear and somewhat simple language was important to Coleridge in constructing his poetry and he uses it artfully to create the beautiful and powerful scenes of the poem.