Short answers, pulled from the story.
The Livonian War, fought from 1558 to 1583, was a struggle for control of Old Livonia in present-day Estonia and Latvia. The Tsardom of Russia fought a varying coalition of the Dano-Norwegian Realm, the Kingdom of Sweden, and the union of the Grand Duchy of Lithuania and the Kingdom of Poland.
Ivan IV invaded Livonia on the 22nd of January 1558, beginning the war. He regarded the Livonian Confederation's approach to Poland-Lithuania for protection under the Treaty of Pozvol of September 1557 as cause for war.
Russia lost. Under the 1582 Truce of Jam Zapolski, Russia surrendered all its Livonian holdings and the city of Dorpat to the Polish-Lithuanian Commonwealth and conceded Polotsk. The 1583 Truce of Plussa with Sweden left Narva, Ivangorod, and most of Ingria under Swedish control.
Stephen Báthory was the Transylvanian prince who became King of Poland and Grand Duke of Lithuania in 1576. He turned the tide of the war against Ivan IV, taking Polotsk on the 30th of August 1579 and Velikie Luki on the 5th of September 1580, and besieging Pskov in 1581.
The Kingdom of Livonia was a Russian vassal state under Magnus of Holstein, who was declared its monarch in 1570. Magnus, the brother of Frederick II of Denmark-Norway, defected from Ivan IV in 1576, and the state nominally existed until that defection.
Livonia was the decentralised and religiously divided Livonian Confederation, with a weak administration, persistent rivalries between the Archbishop of Riga and the Order's Landmeister, and no powerful defences or outside support. Robert I. Frost wrote that Livonia was in no state to resist an attack.