JOEL - The 6th commandment says:

"Thou shalt not kill" (Exodus 20:13)

The Hebrew word "Ratsach", translated to the word "kill" in the Bible means more correctly to "murder" or "slay" a human being.
The Old Testament says:

"he that killeth any man shall surely be put to death." (Lev. 24: 17, See also Ex. 21: 12 and Gen. 9: 6)

Refering to the Ten Commandments Jesus said:

"Thou shalt do no murder, Thou shalt not commit adultery, Thou shalt not steal, Thou shalt not bear false witness," (Matt 19:18)

When considering all these scriptures it is clear that the commandments are intended to refer to how man is to treat other humans; they are obviously not intended to apply to animals.
However, we are taught to have a respect for all creatures God has created. President Brigham Young once said:

“The more kind we are to our animals, the more will peace increase, and the savage nature of the brute creation vanish away” (Teachings of Presidents of the Church: Brigham Young, 1997, 333).

The Prophet Joseph Smith and some other men were camping on the bank of a small river. As the Prophet pitched his tent, he found three prairie rattlesnakes. The other men wanted to kill the snakes, but the Prophet stopped them and told them not to hurt the snakes. The men picked the snakes up with sticks and carefully carried them to the other side of the river. The Prophet told the other men that they should not kill any snakes, birds, or other animals while they were on their journey unless they needed them for food. (See History of the Church, 2:71–72; see also Spencer W. Kimball, Ensign, Nov. 1978, p. 45.)