Grout is generally porous and white or light colored and can get dirty or discolored quickly. Get it clean with a little chemistry and a bit of elbow grease.
First off, you'll want to use an alkaline cleaner. Grout is primarily cement and can be dissolved by acids. Over time, an acidic cleaner may require you to re-grout sooner than intended.
How do you know if your cleaner will do the trick? Typically the product label will tell you the surfaces it works on. Or, if it's says it's meant for bathrooms and cleaning tile, it should work on grout as well.
Spray the product on and then wait for it to work. This will make it easier to clean, using a brush or clean cloth. Then rinse and let dry.
Other alkaline cleaning options:
Bleach - Clean with a solution 3/4 cup of liquid household bleach to 1 gallon of water or a mildew stain remover, applied carefully with a soft-bristled brush (an old toothbrush can be a good option). Rinse thoroughly.
Baking soda - Mix with a bit of water to form a paste, then scrub with a clean cloth or soft-bristled brush. Rinse with warm water and let dry.