Lowering (or lower) springs and coilovers are both parts of a car's suspension. A coilover is a general accessory in suspension, while a lower spring is a specialized piece of suspension generally used for racing. Two kinds of coilovers exist: the standard coilovers found in most cars and performance coilovers, which are height adjustable and allow you to lower your car.
Suspension systems are based on two pieces of equipment: springs and shock absorbers. Springs are what absorb the initial shock of bumps in the road, while shock absorbers are hydraulic pistons that take the remainder of the shock from the spring.
Coilovers are the most commonly used form of shock absorber because they take up so little space. They are a combined spring and shock aborber; indeed, this is where the name comes from -- the spring is "coiled over" the shock absorber. Standard coilovers appear in most cars, while height-adjustable coilovers allow you to use coilovers while also lowering your car.
Lower springs are a specialized spring specifically designed to be small. A large spring allows the wheel to move a lot in a vertical manner. This is ideal for general driving, but it is far from ideal for racing, where the conditions are smoother and handling is more important. Lower springs are generally used in racing cars to get a little extra handling at the cost of smoothness.
