What are Microaggressions?
Microaggressions are small things that hurt in everyday life. Something which, by itself, would be minor, but they pile up over time and can cause real harm. It's kind of like a paper-cut. If you just got one paper-cut, it would be no big deal. If you got ten paper-cuts every day, it would really start to hurt. By the tenth paper-cut that day, you might react a little strongly. Other people might say "wow, they sure are overreacting to that paper cut. It's just one little cut." They don't realize you're covered in them.
People who are marginalized popover available do get many of these papercuts every day! For a woman, hearing one sexist comment might be easy to brush off. But because she may hear them on a daily basis, it comes to hurt much more each time. What makes microaggressions aggressions and not just regular rudeness is that they target core parts of one's identity. Particularly, they target a part of one's identity where they are marginalized. Microaggressions can be verbal tooltip available, non-verbal tooltip available, or a part of the environment tooltip available.
In his book: "Microaggressions in Everyday Life: Race, Gender, and Sexual Orientation" Dr. Derald Wing Sue provides this personal example of a microaggression.
I (Derald Wing Sue, an Asian American) recently traveled with an African American colleague on a plane. [...] As the attendant was about to close the hatch, three White men in suits entered the plane, were informed they could sit anywhere, and promptly seated themselves in front of us. Just before take-off, the attendant proceeded to close all overhead compartments and seemed to scan the plane with her eyes. At that point she approached us, leaned over, interrupted our conversation, and asked if we would mind moving to the back of the plane. She indicated that she needed to distribute weight on the plane evenly.
Both of us (passengers of color) had similar negative reactions. First, balancing the weight on the plane seemed reasonable, but why were we being singled out? After all, we had boarded first and the three White men were the last passengers to arrive. Why were they not being asked to move? Were we being singled out because of our race? Was this just a random event with no racial overtones? Were we being oversensitive and petty? Although we complied by moving to the back of the plane, both of us felt resentment, irritation, and anger. In light of our everyday racial experiences, we both came to the same conclusion: The flight attendant had treated us like second-class citizens because of our race. But this incident did not end there. While I kept telling myself to drop the matter, I could feel my blood pressure rising, heart beating faster, and face flush with anger. When the attendant walked back to make sure our seat belts were fastened, I could not contain my anger any longer. Struggling to control myself, I said to her in a forced calm voice: “Did you know that you asked two passengers of color to step to the rear of the ‘bus’”? For a few seconds she said nothing but looked at me with a horrified expression. Then she said in a righteously indignant tone, “Well, I have never been accused of that! How dare you? I don’t see color! I only asked you to move to balance the plane. Anyway, I was only trying to give you more space and greater privacy.”
Attempts to explain my perceptions and feelings only generated greater defensiveness from her. For every allegation I made, she seemed to have a rational reason for her actions. Finally, she broke off the conversation and refused to talk about the incident any longer. Were it not for my colleague who validated my [...] reality, I would have left that encounter wondering whether I was correct or incorrect in my perceptions. Never-the-less, for the rest of the flight, I stewed over the incident and it left a sour taste in my mouth 
The above story actually has two examples of microaggressions. The first one is when the flight attendant gave the comfort of the white passengers priority over the comfort of Dr. Sue and his colleague. The second microaggression was when the flight attendant claimed to "not see color" and even claimed she had been trying to help. The first microaggression made Dr. Sue and his colleague feel like second-class citizens. The second made them feel paranoid and wrong to have felt hurt in the first place. This example also mentions the most common coping skill for dealing with microaggressions: Getting validation from peers. By asking his colleague for confirmation of what just happened, Dr. Sue was able to offset some of the paranoia, anxiety, and self-doubt from the microaggression.
Here's another example borrowed from my own life.
One day, I (a white transgender tooltip available woman) was out shopping with a friend of mine (a white transgender man). We entered a small shop selling vintage clothing and were the only ones in the store besides the shop owner (a white cisgender tooltip available man). After about a minute, the shop owner changed the music playing in the store to the Aerosmith song, "Dude (Looks Like a Lady)." My friend and I exchanged looks, but decided it must be a coincidence and kept browsing. The next song to come on was "Lola" by The Kinks, a song whose lyrics tell the tale of a straight man being "tricked" into having sex with a transgender woman. At this point, I was pretty uncomfortable and asked my friend if we could leave the store. My friend still said it must be a coincidence, and insisted on staying until he finished his purchase. I left the store feeling paranoid and freakish.
In the above story, there are once again two examples of microaggressions. The first one is the store clerk deciding to make fun of me by playing particular songs about "men who look like women." The action was intentional on the part of the store owner, and was meant to demean me as a trans women. The second microaggression was on the part of my friend, (the white transgender man) when he dismissed my concerns and insisted on staying in the store. It was completely unintentional on the part of my friend, and may have even been well-intentioned. Despite that, it left me feeling paranoid and irrational. Even though he was my friend, it was still a microaggression. Microaggressions can still come from friends, family, and loved ones.[1-2]
According to the philosopher Judith Butler, microaggressions "put people in their place." A transgender woman might be feeling really a part of her majority-cisgender workplace, but a transmisogynist tooltip available microaggression would remind her that the majority of her co-workers are cisgender and do not see her as an equal. The stress and other negative emotions that follow are referred to by social psychologists as minority stress. Somewhat a misnomer, since marginalized people still can experience minority stress when they are not in the minority. For instance, Latino/Latina people make up 65% of Miami, FL, but are still treated as second-class citizens by the privileged white population, and therefore still experience "minority stress." 
For more examples of microaggressions, and elaboration on what is a microaggression versus a regular aggression, see The Line Between Microaggression & Aggression.
 Derald Wing Sue. (2010). Microaggressions in Everyday Life: Race, Gender, and Sexual Orientation. Hoboken, NJ: John Wiley & Sons, Inc.
 Kevin Nadal. (2013). That’s So Gay: Microaggressions & The LGBT Community. Washington, DC: American Psychological Association.
 Judith Butler. (1997). On Linguistic Vulnerability. In Excitable Speech: A Politics of the Performative (pp. 1–41). New York, NY, USA: Routledge.