What are some connotations of "The Tyger" by William Blake?

William Blake was a metaphysical poet, meaning that his poems often held connotations about the deeper questions of life and the meaning of existence. I like that you have have asked about "connotations" in plural form, for a poem or any piece of literature can have multiple connotations. A connotation is the deeper meaning the reader takes from the work; it does not have to reflect the connotation the writer hoped to impart--as long as it is based on an accurate reading of the text. 

To draw connotations from "The Tyger," it's helpful to compare it to its matching poem. Blake wrote "Songs of Innocence" and "Songs of Experience," two collections of poetry that demonstrated his belief that "without contraries is no progression." Often two separate poems will have the same title, such as "The Chimney Sweeper" or "Holy Thursday." Such pairings represent contrasting views of the same subject, the first viewed from the perspective of "innocence" and the second viewed from the perspective of "experience." "The Tyger" is a song of experience, while its contrary is "The Lamb," a song of innocence. 

"The Lamb" presents the soft side of God and His loving kindness. "The Tyger" shows a Creator who has made a dangerous, ferocious beast and questions, "Did he who made the Lamb make thee?" The tiger is described as possessing "fearful symmetry" with a brain filled with "deadly terrors." It can be taken as the representation of cruelty and violence, or, in a word, Evil. The question the poem poses, then, is: Did God create Evil? If not, where did it come from? If he did, then perhaps we need to rethink who God is. Thus, one connotation is that as a person matures and grapples with the reality of evil in the world, he or she begins to question his or her childhood faith in God.

However, those who hold to an orthodox Christian worldview can find a different connotation in the poem. Christians are very comfortable with the way "The Lamb" presents God, noting that "He is called by thy name," meaning that Jesus Christ was known as "the Lamb of God who takes away the sin of the world" (John 1:29). But the Bible also teaches that "our God is a consuming fire" (Hebrews 12:29). In Romans 11:22, the Apostle Paul urges Christians to "consider therefore the kindness and sternness of God." Christians see in these two poems a validation of those two sides of God, and they find their faith strengthened rather than weakened by the comparison. 

Removing the question of religion, one could also take as a connotation the idea of maturing or gaining experience. An adult needs to be aware of the dangers of the world and leave naive notions of the innate goodness of mankind and the world behind. This would support the view of Naturalism: that a cruel and uncaring natural world over which one has no control wields great, even ultimate, power in this life. 

The penetrating questions posed by "The Tyger" have intrigued thoughtful people for over two centuries, encouraging us to explore the topics of God, Nature, good, and Evil.