For practical painting purposes, there are three white pigments. It's useful to know their properties, because they each have their uses.
Lead white (aka flake white, cremnitz white, ceruse) is the traditional white that's been in use since ancient times. It is basic lead carbonate (older versions of the pigment had some other lead compounds mixed in).
For the most part, lead white is available only in oil paint. It is of moderate opacity and has a slightly warm tone. Lead white forms a flexible and permanent paint film. In oil paint it dries very quickly. I find that lead white makes a great general mixing white, less overpowering than titanium and with more strength than zinc. Some people are afraid of lead white because they've heard so many bad things on the news about lead house paint. The problems that occur with leaded house paint are not relevant to painting unless you plan to allow children to eat your paintings (something I would strongly caution against). Lead white is, in fact, a bad thing to ingest (especially for kids) and anyone using it should be careful. But it isn't radioactive and it doesn't penetrate skin, so a few simple precautions (which should be used when painting no matter what pigments you work with) are what's needed to be safe with lead white artist's paint.
Over hundreds of years, lead white becomes more transparent. This transparency is the reason why you can often see the ghostly images of changes the artist made as the painting developed (these are called pentimenti—"repentances"). It is also why early Renaissance Italian egg tempera paintins often have green flesh tones. The usual procedure was to paint flesh areas with green earth, followed by a mixed dark dull tone called a verdaccio, followed by red on the parts of the face that have lots of blood, such as cheeks, ears, and the nose. After that, a pink made from lead white mixed with vermillion was applied. As the lead white becomes more transparent, the green and red underlayers are revealed. Modern painters who like to use lead white (and perhaps vainly think that their work will be treasured for hundreds of years) often use mixtures of lead and zinc or titanium to mitigate this transparency effect.
Titanium white is the strongest available white pigment. It is very opaque and slightly cool in tone. Because it is such a strong tinter, titanium can be hard to work with, particularly with darker colors. They become "chalky," losing their chroma rapidly with the addition of even a little titanium white. There are ways to compensate for this, but I generally use titanium white only when I really need heavy duty opacity. Titanium is a slow dryer in oil, which is usually another strike against it in my book, since I like to paint in layers.
Zinc white is the least opaque white of the available pigments. That makes it great for delicate value adjustments, especially with dark colors that are easily thrown off by just a tad too much white. It's also good for glazing or scumbling in multi-layered painting approaches in which underlayers are allowed to show through upper layers. In watercolor and gouache, zinc white is sometimes labelled "Chinese white." In acrylic, it's sometimes called "mixing white."
Mixes of these colors can be useful. For example, Williamsburg makes an excellent titanium-zinc blend that's opaque but less overpowering than pure titanium. Doak's flake #1c is a lead white and zinc blend that is great as a general mixing white (and less likely to suffer from eventual transparency than pure flake). I often have both of these on my palette.
In egg tempera, I often use a 50/50 blend of titanium and zinc. This gives an effect similar to lead white, without having to deal with toxic pigment in powder form. The smallest addition of yellow ochre gives it the warm tone of lead white as well.