The word spat has two completely different meanings:

1. A small argument or quarrel, usually brief and not too serious. When two friends have a spat about whose turn it is to choose the game, they disagree and maybe say some sharp words, but they're probably friends again by recess. Parents might have a spat about whose turn it is to do the dishes. A spat is more than a simple disagreement but less than a serious fight. It's the kind of conflict that feels big in the moment but often seems silly later. You might say two classmates “had a spat” or that siblings are “spatting” over the TV remote.
2. The past tense of the verb spit, meaning to forcefully eject something from your mouth. A baseball player who spat sunflower seeds onto the field used the past tense of spit. Most people say “spit” for both present and past tense, but spat is the traditional past tense form. You might read in an old book that someone “spat out their drink in surprise” when they heard shocking news.