The Mandalorian-Jedi War mentioned in The Mandalorian was fought thousands of years before the events of the main Star Wars saga, and it came to define the Mandalorians as a race, not to mention their hatred of Jedi. Originally introduced in the 2003 video game Star Wars: Knights of the Old Republic, the Mandalorian Wars were used to set up the story and explain what happened to some of the main characters, namely Revan and Malak, who were once Jedi but later became Sith.
Just short of 4000 BBY, Mandalore the Ultimate launched a large-scale war against the Republic - partly under the influence of the Sith - and sought to make the Mandalorians the rulers of the galaxy. Mandalore's Neo-Crusaders went mostly unmatched despite invading the Republic's territory and destroying countless cities along the way. The Republic just wasn't strong enough to beat the Mandalorians, and unlike what happened in the Clone Wars, the Jedi weren't allowed to fight either - that is until Revan and Malak formed their own Jedi army. They joined the fight and pushed the Mandalorians back until there was a final confrontation on Malachor V.
Revan killed Mandalore the Ultimate on that planet and consequently ended the Mandalorian-Jedi War. But unfortunately, most of that story has been relegated to Star Wars Legends given that they happened prior to Disney's acquisition of Lucasfilm. In the new Star Wars canon, the Mandalorian-Jedi War was fought by Mandalore the Great, who challenged the Jedi and occasionally the Republic itself. The war ended on the Mandalorian's homeworld Mandalore, with most of the surface destroyed in the conflict.
So far, only a handful of events from the Legends version of the Mandalorian-Jedi war have been mentioned in canon, such as the crusades, the Devastation of Ubduria, and the fact that the Mandalorians were led by a ruler - though his name was Mandalore the Great instead of Mandalore the Ultimate. A significant portion of those stories was explored in Clone Wars season 2, but with that series now over, it seems the torch has been passed on to The Mandalorian.
The Mandalorian season 2 trailer uses dialogue from the season 1 finale in which the Armorer tells Din Djarin about their race's ancient war with the Jedi. While the war's mention in season 1 was merely a tease, it seems to be a focus for season 2's story - given that the Mandalorian himself is searching for Baby Yoda's species as well as the Jedi. And with Ahsoka reportedly appearing in season 2, she may be the key to unlocking the mystery behind who Baby Yoda is, and audiences could in turn learn more about the Mandalorian-Jedi War. After all, she knows the Mandalorians quite well herself.