Get the Shine Back on a Laminate Floor

Most often, the shine is just hidden under a haze caused by using incorrect cleaning methods.

Steps

Dampen a rag in plain white vinegar and rub the floor until the shine is restored. If this restores the shine, you'll just need to clean the floor properly. If the shine has worn down due to wear, you'll have to use a laminate floor shine restoration product.
First, sweep or vacuum the floor to be cleaned. Any particles of dirt could scratch the floor if it's ground in with a mop.
Using a bucket of hot water, add 3/4 to 1 cup of vinegar, depending on the size of the floor.
Dip the mop into the water and vinegar mixture, wring as tightly as possible so that the mop is damp, not dripping wet.
Rinse the mop often in the water/vinegar to remove any residue from the floor.

If there's a heavy haze on the floor, you may need to repeat the above a few times to restore the shine.
If you see any spots/stains on the floor, simply wipe them up with a damp cloth/paper towel. If the stain is oily/greasy, dampen the cloth with vinegar to remove the oily stain.
You should never use an all-purpose cleaner (the type used to clean vinyl or ceramic floors) on laminate floors. Unless the floor is really dirty, most of the time you'll just need to sweep/vacuum and mop with plain hot water.

If you need to use a shine restorer, make sure you purchase an item specially designed for laminate floors; don't use wax meant for tile, vinyl or no-wax floors.
Never leave water standing on a laminate floor. Water could seep between the cracks and cause the floor to swell and warp. Always wring the mop as tightly as possible.