The difference here is all about the indefinite vs. the definite articles.
The definite article is most commonly used in combination with a noun or phrase that has already been mentioned or is known by both hearer and speaker through context.
So, about these sentences:
- The boys who are 16 years old.
- Boys who are 16 years old.
In the first sentence, a specific group of boys are implied. For example, if you were talking about a party and then mention the boys who are 16 years old, you mean all boys who were at that party and are 16 years old. In the second sentence you're talking about boys in general.
There is a reason why this sentence is not correct:
Boy who is 16 years old.
In the plural, it works without an article. This is because actually you do use an article! We call it the zero article. It is the plural form of the indefinite article. In each of the following sentences we have an article, the second one is just the zero article that you do not actually see, because it is not written:
Note that there are more uses for the definite article than the one I just gave you, such as indicating unique nouns (the sun, the King, etc.), time periods (during the night, in the sixties).
Also, it may not always be clear when to use the definite or indefinite article. Have a look at this: Boys who are 16 years old. vs The boys who I saw yesterday. The first sentence is a more 'general' sentence, it is much less restrictive than the second where the hearer can understand which boys are meant. You will have to ask yourself how restrictive the postmodifier is. Does it allow the hearer to specifically identify who the speaker is talking about? If not, you should use the indefinite article such as with boys who are 16 years old. Though boys is being restricted by the postmodifier, it still expresses a rather 'general' group of people. Oppose this to the boys who I saw yesterday and you can tell there is a difference. This group of boys is much more specified and allows the hearer to form a specific interpretation of the people being talked about.
You can read more about the indefinite vs definite articles here: Indefinite and Definite Articles