1. Semicolons - These tricky little suckers are used to separate independent clauses with closely related ideas. Often, people will use them as commas to interject a sentence, which is incorrect. 

Incorrect: Many girls prefer drinking Starbucks; while I prefer The Aussie Bean.
Correct: Some girls prefer drinking Starbucks on a daily basis; I prefer The Aussie Bean.

2. Was vs. Were - When it comes to the subjunctive or indicative mood, these two words can get a little mixed up. In the subjunctive mood (hypothetical situation or contrary to reality), use "were". In the indicative mood (statements or questions), use "was".

Were: If Kelley were a Zeta Beta Zeta, I'd want her as my big. 
Was: When I was going through recruitment, I really liked talking to Kelley.

3. Farther vs. Further - Though the use of these words has become accepted as interchangeable, farther is used more often to describe physical distances, whereas further is used to describe a figurative distance

Farther: Today, the lacrosse team ran farther in their warm up routine than they did yesterday.
Further: All the running the lacrosse team has done is helping them further their stamina.

4. Anyway vs Anyways - "Anyway" is an adverb and "anyways" is used for the same meaning, but has a more colloquial effect. While it is not necessarily incorrect, "anyways" should be avoided in an academic or professional environment.

Anyway: President Doti, I'm sorry we couldn't speak longer, but it was great talking to you anyway.
Anyways: Anyways, it was so nice getting lunch! We should do this again soon.

5. But vs. However - These two are on the list because many people use "but" when they could use "however", especially when writing counterarguments.

Incorrect: Sami wanted to go to a party on Sunday. But she decided not to because she had a midterm on Monday.
Correct: Sami wanted to go to a party on Sunday; however, after considering the consequences, she decided to stay home and study for her midterm.

6. Oxford Comma - This comma is placed before the conjunction in a series of three or more. Whether or not the Oxford comma (or serial comma) should be used is widely debated among different style guides and languages. In some situations, it creates confusion; in other situations, it eliminates confusion. Whether you use this comma is up to your discretion.

Without the Oxford comma: I walked through the Student Union and saw two models, Jim Doti and Jerry Price. (This implies that Jim and Jerry are the two models I saw)
With the Oxford comma: I walked through the Student Union and saw two models, Jim Doti, and Jerry Price. (This could imply that I saw Jim and Jerry accompanied by two models)

7. Good vs. Well - Responding to "How are you?" with "I'm good" has become so common and automatic that many people aren't even aware it's incorrect. "Good" is an adjective and "well" is an adverb, meaning that "good" describes a noun, while "well" describes a verb.

Good: I think we did a good job with our philanthropy event because we raised a lot of money for our charity organization!
Well: How did the philanthropy event go? I think it went well because a lot of people attended and donated to the cause!

8. Effect vs. Affect - A lot of people admit that even though they try to remember the difference between these two, they still get confused.  Effect a noun that is used when something happens due to a cause (remember "cause and effect"). Affect is a verb that means "to change" or "to influence" 

Effect: I've developed great social and leadership skills as an effect of joining Greek Life. (Cause=joining Greek Life. Effect=social and leadership skills.)
Affect: Being a part of Greek Life has affected me in such positive ways and has inspired me to be a better person. (See how affected and inspired are both verbs?)

9. We vs. Us - The type of sentence these two get mixed up in isn't as common, but it's still helpful to know what's correct in case you decide to use it in a paper.

Incorrect: Us Zeta Beta Zeta girls would like to cordially invite you to our sisterhood event! (This is incorrect because if you took out "Zeta Beta Zeta girls", you would be left with "Us would like to cordially invite you...")
Correct: We Zeta Beta Zeta girls would like to cordially invite you to our sisterhood event! (See how "We would like to cordially invite you..." can stand on its own?)

10. Passed vs. Past - These two sound exactly the same, so it's a bit challenging to remember what means what all the time. Passed is only ever used when referring to the past tense of "to pass". Past can be used as a noun meaning "before the present" or as an adjective meaning "finished" or "completed". 

Passed: Only three days have passed since formal, but I still have constant withdrawals.
Past: Our formal on The Queen Mary is better than it has been in years past.  

11. Adverbs - A lot of people know that adverbs are the words that end in -ly and that they modify the meanings of verbs, adjectives, prepositions, conjunctions, and other adverbs. However, there are too many casual conversations that eliminate the use of adverbs where they are needed.

Incorrect: She drank her Pumpkin Spice Latte really slow in order to savor its deliciousness.
Correct: She drank her Pumpkin Spice Latte really slowly in order to savor its deliciousness. (If you rearranged the sentence, "She slowly drank..." would make more sense than "She slow drank...")

As a grammar enthusiast (and as an English major), I strive to always improve the way I speak and the way I write. Hopefully the knowledge I am passing on also enables you to improve the way you speak, which can help you sound more professional in the appropriate setting, and improve the way you write, which can help you score more grammar points on your papers! Of course, these are only eleven grammar rules/tips out of an endless list, so here's a great grammar website that can help you with any other questions you may have!