Articles are perhaps the most persistent problems for writers whose first language doesn't use articles. Other writers are used to languages in which every noun requires an article, and it is unclear to them when articles should be omitted. On the other hand, a native speaker (even a young one) will never have trouble with articles: we know at some fundamental level when to use or to omit "the," "a," or "an."
To understand how articles are used, a writer must be able to distinguish whether the noun an article modifies is definite (specific) or indefinite (general). This depends on context. An indefinite noun represents something that is one out of many (its particular identity is unknown): a dog, some students, some air. A definite noun is something of which we know the particular identity: the dog, the students, the air. All definite nouns come with the definite article: THE. Indefinite nouns come with an indefinite article (A or AN) if they are singular and countable (a dog). Indefinite nouns that are plural (students) or uncountable (water) do not take an article.
Since usage is very context-specific, the difficulty lies in determining whether a noun is definite or indefinite and why this is so. Consider: "Mr. Kim, President of Dartmouth College, urged the students to find a passion that would inspire them to great deeds in the benefit of humanity." It's not easy to explain the article usage in a sentence like this! After all, President Kim could also "urge students" and still be grammatically correct! The choice he makes depends on his intent.
Making the problem even more difficult to explain is that some article use is idiomatic, or requires a lot of grammatical analysis in order to be understood. For example, why do we invite someone out to dinner, not out to the dinner, or adinner? (Though we will always invite them out for a meal, not simply meal.)
You will do a lot of intellectual sweating attempting to help writers with articles, and you will be tempted to pass off most of what is hard to explain as idiomatic. Resist this temptation! Perhaps the usage is idiomatic, but an attempt to explain and to understand the finer points of grammar can be useful for your tutees - and for you as well.