Everyone that has tile in their home would like for the grout between the tiles to stay clean and beautiful forever. Unfortunately it just doesn't happen that way. Here are a few ideas for cleaning that may help to bring back some of that beauty.
If stains are light you may be able to use the edge of a piece of sandpaper to lighten it up. Make sure you are not sanding your grout line too deeply unless you plant to replace some of it. Sometimes erasers work or the white sole of certain tennis shoes. You know, the same ones that seem to erase black scuff marks from linoleum. Do a test area if you can, especially if you are cleaning colored grout.
If that doesn't work try a soft bristle toothbrush with neutral grout cleaner. Neutral meaning its PH level. If the dirt and grime is just too much you may have to resort to removing and replacing some or all of the grout.
Do not use vinegar or bleach. These products are actually too corrosive and can damage the structure of the grout itself. The grout will crumble out earler than it would have before using the bleach. Also if you are spot cleaning and your grout gets too white, it's not going to match the part you didn't clean.
You might have a little fun with a Dremel rotary tool. There are bits designed especially for this type of job. These are fun tools and this is an easy way to clean tile grout. This goes fast and is really easy. I would suggest wearing an inexpensive paper mask if you do this just to keep the dust out of your lungs.
If you do end up having to remove the old grout, one way is to use a grout saw. This tool is specially designed for removal of hardened grout. Be prepared to sweep or vacuum up grout dust so you don't breathe it in and track it all over the house. A grout saw is not expensive and all you need to is follow manufacturer's directions.
Once you have your grout looking like you want it use a grout sealer. It should protect the grout from further staining and grime buildup for several years.
If this is tile flooring and you have gone to this much trouble to get your grout nice looking this would be a good time to look at ways to prevent having to redo this in the near future. You will want to keep the tile floor swept on a regular basis so dirt particles are less like to get ground in. Wipe up any spills immediately. If you use a mop to clean your tile floor make sure you rinse it often so it is not leaving behind more stained water.