I have a beautiful african violet, but the new small leaves in the center are turning brown on the edges. I broke them off and the same is happening to the new ones.
Also, on other plants, what does it mean when the leaves turn yellow?

Make sure your Av is free from any pest like cyclamen mites, they are tiny, so you 'll probably want to use some kind of magnification.You don't want any water sitting on any parts of the plant, especially if it gets any little bit of sun at all. Salt build up from over-fertilizing can also cause brown edges.

If you mean by other plants yellowing, other than Avs, yellowing leaves can be caused by over-watering, or a light change, light being too low or even seasonal. From bright summer light to dull winter, from dull winter light back to brighter summer can cause yellowing with some plants on some leaves, nothing your doing in that case. My pothos does that. A few turn yellow in late fall/winter, then the plant is fine, then a few more yellow leaves in March/April when we get higher light here in the north east. Older leaves and some bottom leaves can yellow which can be normal.