Rate = k[A][B] and Rate = k[A]2 are both second order reaction. In the first case, A and B are both first order and in the second, A is second order.
If a reactant is first order, then when its concentration is doubled, the rate of the reaction doubles; if the concentration is tripled, the rate triples, etc.
If a reactant is second order, then when its concentration is doubled, the rate of the reaction quadruples (22 = 4); if the concentration is tripled, the rate increases by a factor of 9 (32 = 9)
(If a reactant is third order, then when its concentration is doubled, the rate of reaction increases by a factor of 8 (23 = 8), etc.)
If a reactant is 0 order, then changing its concentration does not affect the rate of reaction.