If you are trying to educate yourself regarding the SpaceX vs Blue Origin conflict you first need to understand what are these two companies basically:
Blue Origin is a private aerospace manufacturing company also providing sub-orbital spaceflights. It was founded and is owned by Jeff Bezos. The headquarters of the company is located in Kent, Washington.
SpaceX is owned and founded by Elon Musk. It is also an aerospace company with its headquarters located in Hawthorne, California. The vision behind the company was to cut down costs of space traveling and colonize planet Mars.

SpaceX and Blue Origin Rivalry Timeline
Now you must be wondering what exactly happened and why do people discuss it so often? Both the companies have conflicts against each other for a very long time. Just like a casual industrial rivalry. However, the owners are quite powerful which make it an interesting case. Following is a summarized version of the years-long rivalry between two major companies of the world.
Blue Origin and Space X decide their press releases and other important events and occasions announcements in a completely competitive manner.
2013
For the Apollo moon missions, both SpaceX and Blue Origin fought to win the lease LC-39A, the rocket launch platform. SpaceX got the lease, back in the year 2013, but Blue Origin sued them. Right now, the lease is still with SpaceX. For the time being, Blue Origin got SLC-36 on rent.
2014
Later, SpaceX also filed a case against Blue Origin to cancel their permit of landing rockets on the ships. In 2014, they won the court case. Since 2014, the company is trying to land rockets at sea. They were successful in 2016, which was way before Blue Origin’s rocket landing platform.
2015
In 2015, SpaceX and Blue Origin got into Twitter beef regarding the meaning of landed rocket, used rocket, and space rocket. It was after when New Shepard successfully completed his voyage and landed back on Earth. Before this scenario, SpaceX tried to launch and land their rocket Grasshopper quite a few times but all without reaching space. However, it managed to land a Falcon 9 rocket first stage. It was later used to launch a satellite into orbit. All of this triggered many more twitter fights in 2016.
2016
Later in 2016, Blue Origin announced New Glenn. It was direct competition with Falcon Heavy of SpaceX. However, New Glenn was a bigger rocket with a lower payload so that was a win-win situation.

At the International Astronautical Congress, Mexico in 2016 president of Blue Origin, Rob Meyerson explained the vision of Bezos that was outlined previously in the New Glenn announcement. Moreover, SpaceX revealed Interplanetary Transport System at the same meeting against which Blue Origin unveiled the New Armstrong. Both of these inventions have similar purposes and functioning. Like we mentioned earlier both the companies have a severe rivalry against each other. All the important reveals and announcements are carefully planned to compete with the opponent at the right time.

2021
Recently, in April 2021 SpaceX again defeated Blue Origin for a 2.9 billion dollar contract. The purpose of the mentioned contract was to invent the lunar lander for the Artemis Program of NASA.
The conflict between SpaceX vs Blue Origin has been going on since 2013 and it feels like it is here to stay for a while. Since it will be a huge achievement, whoever manages to win the billionaire’s race and get to MARS. Therefore, the fight is much harder with much more time, effort, and energy put into than needed. All we can do is wait and watch to find out who wins and what’s the future of this entire fiasco.