In this paper, we study Maker-Breaker games on the random hypergraph $H_{n,s,p}$, obtained from the complete $s$-graph by keeping every edge independently with probability $p$. We determine the threshold probability for the property of Maker winning the game as a function of $s$, the uniformity of the underlying hypergraph, as well as $m$, $b$, the number of vertices that Maker and Breaker are respectively allowed to pick each turn. In addition, we show that depending on those $m,b,s$, there are two types of thresholds: either being Maker-win is a local property and the threshold is weak, or it is related to global properties of the random hypergraph and the threshold is semi-sharp. We conjecture that in the latter case, the threshold is actually sharp.