After a lengthy discussion with a talented user, BillJ, on this site, I have gotten a pretty good look at complements and how they function, so I have followed through with the suggestion to post an answer to my own question:
When examining complements, I find it best to look at structures syntactically, more so than semantically, as can be alluded to by the definition of "complement" -- "A syntactic element seen as completing the construction of another element," according to Matthews in The Concise Oxford Dictionary of Linguistics. Furthermore, as discovered, complements are best described as obligatory elements licensed by some other element such as a verb.
Being able to differentiate between complements and adjuncts (or modifiers) is obviously one of the keys to defining complements. If you remove a complement from a sentence, it becomes ungrammatical or at least lacking of essential meaning.
From an example provided from BillJ,
"Sue used the cheese." (Obligatory, Complement)
"Sue ate the cheese." (Modifier)
In the first sentence, the verb "used" obviously calls for an object--or at least something to fulfill the action, as "Sue used" is not complete without "the cheese". The second sentence, however, does not need the object, "the cheese", because "Sue ate" is perfectly acceptable.
But, this is sometimes a lot more complicated, and it can be hard to understand what exactly is obligatory. Additionally, this is why it is important to focus on the sentence in both a syntactic viewpoint and in one where you recognize the call for licensing that certain elements possess. I am just now starting to get the hang of distinguishing between complements and modifiers.
Moving on, it is also important to note that complements can function in many different layers of sentence structures. One thing I did discover was that relative clauses and pronouns are indeed modifiers for structures as a whole. But, within these clauses are complements. For example, the sentence "The walk which I had taken my dogs on was wonderful." Within this sentence, we have the relative pronoun "which" introducing the relative clause "which I had taken my dogs on". This relative item, in regards to the rest of the overall sentence, is not a complement. It is a modifier. However, if you look closer, you will see that "which" does indeed perform the role of complement...in the relative clause to the preposition "on".
Another example would be the sentence, "The boy who I loved cared nothing for me." In this sentence, the relative clause headed by the relative pronoun "who" is again a modifier, but, again, the relative pronoun can be said to be acting as complement within its own clause, the relative clause, to the verb "loved", as an object.
One might think that the mentioning of these relative clauses is not beneficial to the understand of the term "complement". However, It is essentially what allowed me to understand complements, as I was able to recognize how they operate.