Salaams,
Firstly, I'd like to wish all Muslims a Happy New Year.
Now, let's get to the "meat and potatoes" of this topic. The Massacre of Karbala was a massacre that happened on the 10th day of Muharram in the year 61 AH. It was a battle between Hussein b. Ali (ra), his family and companions, and the forces of a man known as Umar b. Sa'ad who was taking orders from the governor of Kufa, Ubayd Allah b. Ziyad, who was appointed by Yazeed b. Muawiyah. Umar b. Sa'ad won the battle and massacred Hussein b. Ali (ra) and all of his men.
This battle is primarily remembered by our unhappily separated brethren in the Shiite sect whom commemorate the battle every year by gathering in Husseiniyyahs (the Shiite version of a Musalla), singing poetry, listening to lectures, and beating their chests. They continue this period of mourning for a grand total of 50 days from the 1st day of Muharram to the 20th day of Safar (known as Arba'een "the fortieth"). The first 9 days are basically practice and the real mourning begins on the 10th of Muharram (hence why the 20th day of Safar is known as Arba'een and not Khamsoon).
Our friends in the Shiite sect have a very unique reading of the Massacre of Karbala. Those of us who are part of the jama'ah generally view the massacre as a terrible event in history where an overzealous general (Umar b. Sa'ad) committed horrible crimes at the direction of a tyrannical governor (Ubayd Allah b. Ziyad) who was placed in his position by an unrepentant King (Yazeed b. Muawiyah). The Shiites, however, completely disregard the culpability of Umar b. Sa'ad and Ubayd Allah b. Ziyad. Their names are very rarely mentioned in any discussion about what led up to the Massacre and who was really at fault. Rather, they view Hussein b. Ali (ra) as a perfect being who was carrying out absolutely righteousness against a Satan-on-Earth (Yazeed b. Muawiyah). To the Shiites, there is absolutely nobody to blame except for Yazeed, even though there was no way for him to even know about the Massacre until weeks after it occurred (unfortunately, there was no Facebook or Whatsapp in the 8th century, they had to actually ride horses all the way to Damascus to talk to Yazeed). Additionally, they believe that Hussein b. Ali (ra) would want them to beat their chests and occasionally slice their heads open every year in His honor, instead of fighting for the principles that He (ra) fought for.
It is at this point that I must admit that I am not the most eloquent person that could be having this talk. I am an uneducated layman and I think it would be best if we had a true scholar presenting these lessons. Thankfully, there is a real scholar who has already done the great work of recording a 2 hour long majlis on this topic. He discusses the Massacre of Karbala from beginning to end. He talks about all of the events leading up to the Massacre as well as discussing culpability and who is most at fault. He also discusses the differences in opinion between the Muslims and the Shiites in regards to this horrible Massacre.
I present to you, Shaykh Yasir Qadhi:
https://www.youtube.com/watch?v=IN1tPyHAgAo