When the floor is dry you are ready to grout. This step finishes the look. The grout is applied to the floor and a rubber trowel is used to push squeeze the grout in behind the tile. It should go in and form a reverse T behind the tile to hold the grout seam in place. This takes a good wrist action and effort. Do not just smooth it out you need to push it in these seams. Work in small areas of about 1 square foot at a time.
I tend to grout in a 2×2 square area then clean up that portion before moving on.
Each time you pull the sponge across the tile the grout line gets thinner. The look you want is designed by the person doing the grout. I never delegate this step. Keep changing your water and use clean water. Average floor might take 3-4 clean buckets of water.
Note: some tiles have a surface that will hold the grout. Slate for example, you can apply a quick release solution to the tile in advance if you suspect the tile will hold the grout. Ask your retailer for his recommendation. If you have any areas that don’t match, relax you have about 1 hour working time. With a wet sponge, thin the line by continuing to stroke it thinning the line with each pass. When all finished the floor should be neat but will dry to have a light dusting of powder on it.
This is removed by rag or if it is stubborn use a green scrounge from the kitchen supply store. On large floors I had used my automotive orbital buffer to speed up the process. If you wet the floor the dust just keeps coming back lighter and lighter.
Given that your finished look is created by your sponge. I recommend you purchase a good sponge while in the tile store. The sponges they sell are worth every cent. One final note, in high traffic areas you might want to seal the grout to keep the color the way it was. Ask your retailer for grout sealer.