This describes how the rate of reaction depends on concentrations of reactants. In our analogy, for zeroth order reactions, this would be like saying "the speed at which ice cream comes out doesn't change no matter how much is left."
Unlike zeroth order, here the reaction rate depends directly on the concentration of one reactant. If we stick with our ice cream analogy, it'd be like if your machine started dispensing slower as it ran out of ice cream.
Half-Life (t1/2): This refers to the time required for half of a substance to undergo some process such as decay or disintegration. For our ice-cream machine, it could represent how long it takes for half of its contents to be dispensed.