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A negative order means that as the concentration of that substance is increased, the rate of the reaction decreases. An example of this would be a product with a negative order. If the concentration of this product is increased, it participates in a reverse reaction, thereby slowing down the reaction rate.
A reaction rate can have a negative partial order with respect to a substance. For example, the conversion of ozone (O3) to oxygen follows the rate equation (rate = k[O3]^2/[O2]) in an excess of oxygen. This corresponds to second order in ozone and order (-1) with respect to oxygen. When a partial order is negative, the overall order is usually considered as undefined, so in above example, the reaction is not described as first order even though the sum of the partial orders is 2 + (-1) = 1, because the rate equation is more complicated than that of a first-order reaction.
Conceptually, a reaction with a negative order means that as you increase the concentration of reactant, the reaction rate actually decreases. This seems to eb an interesting phenomena to me, and I 'm sent sure of any real life examples of it.
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