Today's standard barrel material is Nickel-Tungsten. Tungsten is a very dense barrel material and popular barrel material, so darts with a considerable weight can be made quite slim. The invention of tungsten darts had a big impact on the sport. With the new darts it was possible to achieve much better grouping than before, and the averages in pro games have improved dramatically. It is not an overstatement when this development is referred to as "the tungsten revolution".
Before the tungsten revolution, darts were mainly made of brass, one of barrel material. Brass darts are still used by some players because they are much cheaper than tungsten darts. But brass barrel material made darts are ancient technology now, so I can't recommend them.
Because of technical difficulties with manufacturing 100% tungsten darts is not available. All "tungsten" darts you can buy are in fact made of tungsten alloys, with Nickel-Tungsten being the most common one. When you buy a dart the amount of tungsten in the alloy is given by a percentage number. So a common "80% Tungsten" dart has 80 percent tungsten and 20 percent Nickel, Copper and/or Ferro. Tungsten darts are available from 80% to over 90%. The more tungsten the slimmer - and the more expensive - the tungsten darts are. If the high-percentage tungsten darts are worth the higher price is an open question. Personally I think that 80% tungsten darts are good enough for almost everyone.
Remember: the only benefit from a high tungsten share is that the dart can be made slimmer than a dart with the same weight and a lower tungsten share. In no way do higher tungsten percentages indicate some sort of overall better quality or better machining.
In recent years a growing number of people has become allergic to Nickel. With Nickel-Tungsten darts being the standard equipment nowadays these people will have problems with their darts. There are a few darts available with different strong tungsten alloys, such as silver tungsten or copper tungsten. However, these darts are very rare. The major dart manufacturers don't supply them. If you are allergic to Nickel you will probably have to look for these alternatives.