Are both of the following correct? If so, when should each be used?
The dog I kicked is sad.
The dog that I kicked is sad.
Both are correct. The relative pronoun "that" can be omitted when it is the object of the verb in the relative clause (the verb "kicked" in your example)
It is a matter of free choice whether you omit it or nor not. Both forms are equally grammatical, equally natural, equally formal.