Every professor has particular grammar or usage "bugaboos". These are some things I see misused most often. I recommend that you review them in one of the suggested style manuals (and avoid them in your writing). Check your paper against this list before turning it in. Good writing is rewriting!
The comma: Commas have very real reasons for their existence. It is frustrating to try to comprehend a sentence that has been lavished with random commas. The primary use of a comma is to separate a dependent clause from the rest of the sentence. (Darn, now you have to look up dependent clauses.) Do not use a comma to join two complete sentences; use a semi-colon for that, or use a period to make two sentences.
The semi-colon: The semi-colon is a useful little fellow; I don't know why it isn't used more often. Apart from its job separating elements of lists, its most useful application is the separation of two independent but related clauses. (Darn, now you have to look up independent clauses. Hint: if you can use a period after a clause, and as long as the following clause can also stand by itself, you can use a semi-colon.
The apostrophe: Apostrophes indicate contractions and possessives. They do not indicate plurals. Possessives get their apostrophes in front of the “s”. Personal possessive pronouns (hers, ours, its, etc.) do not get apostrophes. (Sorry, Virginia, but in my opinion there should be no apostrophe in A Room of One's Own, but it's hard to find anyone to agree with me. Sigh . . .) Plural possessives get their apostrophe after the “s”. Singular nouns that end in an “s” get an apostrophe and an “s”, (as in Chambers's, hint, hint—Chambers' means it belongs to several rooms. Again, there are those who disagree; when you are in their class, do what they want).
The subjunctive: Subjunctives can be tricky; they indicate something to be wished, imagined, or hoped for. Tevya got it right when he said, “If I were a rich man.” If I was a rich man is confusing—and incorrect—because it uses the past tense, “was”, for something that hasn't actually happened.
Agreement: Agreement between parts of a sentence is usually obvious. In "I has a head cold," there is obviously an agreement problem. It might be funny on the I CAN HAS CHEESEBURGER? web site, but “I have a head cold” is correct. Two pronouns can be tricky, however. The statement "Dad asked you and I to take out the trash," sounds formally correct, but it's not. Think of it this way: Dad asked you and Dad asked me, so the correct form would be “Dad asked you and me to take out the trash”. The latest affronts to this are "Her and I" or "Him and me". Really?
Beginning a sentence with a conjunction: I'm sorry, I know many people will disagree with this, some of whom write style manuals, but it just doesn't make sense. Beginning a sentence with “and” is something derived from casual speech and has no place in formal writing. There are more appropriate options. Conjunctions conjoin things; you can't conjoin the beginning of a sentence. Please don't do it when writing a paper for me.
Who and that: A thing should be referred to as that, for a person one uses who. “Ibsen, a playwright who carefully observed human nature, wrote many plays that are considered classics.”
Poof-reading and spelling: Please, for the love of Pete, proof-read your work. Obvious spelling mistakes in a paper written on a computer with spell-checking capability just baffles me. Use a dictionary to be sure you are using the correct word.
Clarity: The ultimate rule in grammar is clarity. Simple sentences that make sense, parts of sentences that agree, the logical and consistent use of parts of speech, and paragraphs that delineate a complete idea are all ways to make your writing understandable to your audience. Anything that stands in the way of the clear expression of an idea and its logical development should be excised from your writing.
I recommend that you put your writing away for a day or two and then reread it, or have someone else read it. If you or your friend can't understand something you have written, I won't be able to either.
Brevity: Prof. Strunk of Cornell said it best in his 1919 The Elements of Style: “Vigorous writing is concise. A sentence should contain no unnecessary words, a paragraph no unnecessary sentences, for the same reason that a drawing should have no unnecessary lines and a machine no unnecessary parts. This requires not that the writer make all of his sentences short, or that he avoid all detail and treat his subjects only in outline, but that every word tell.”
The beauty of those sixty-three words makes me weak in the knees.
(Still my favorite.)