Apache tears, also known as marekanite1, are actually just small nuggets of obsidian. While they have a distinct geological origin, a large portion of their attraction comes from the myth surrounding them, and I suspect that many of the more decorative "Apache tears" are actually just random bits of obsidian that have been polished into beads. If your Apache tears are too pretty, you may have been conned.
Siliceous lava flows may cool into obsidian, but in the presence of water they form instead into perlite, a brittle volcanic glass with so many small fractures that it appears grey or white. It is often the case that perlite contains small round bits of black obsidian glass, usually about 3-5 cm in size, and usually somewhat irregular in shape. We don't fully understand why they form, but it is probably related to cooling rates, the amount of ambient water, and alkalinity2. As the conditions were evidently gradated during their formation (otherwise, the rock would be pure perlite), Apache tears have a pitted, irregular surface with onion-skin fractures.
They are most often polished before sale, but are attractive regardless. They tend to be a smooth black overlain with chalky white, sometimes approaching snowflake obsidian in appearance. If the perlite is completely polished off they look rather like prunes, although some people don't know when to stop and will polish them into boring black marbles indistinguishable from standard obsidian. If you are particularly lucky, you might find some in brown or even brick red, although these forms are nearly always referred to as marekanite rather than tears.
The name marekanite comes from the Marekanka River, Okhotsk, Siberia in Russia,where they were first found. The name Apache tear comes from an Apache legend of Apache warriors that fought the US Cavalry circa the 1870s. When it was clear that they could not win the battle, the warriors rode their horses over a cliff rather than surrender. The tears shed by their families turned to stone. I suspect that it is a later addition to the legend, but it is also said that anyone who carries the Apache Tears will never need to cry again, as the families of the warriors cried enough tears for all who mourn.
Footnotes:
1. There is some confusion in this point; some sources define marekanite as a synonym for Apache tears, some as a synonym for perlite (Oxford Dictionary is the most convincing of the later group). The general consensus appears to be that marekanite is indeed a synonym for Apache tears.
2. Some sources confuse Apache tears with Pele's tears, and claim that they are formed when obsidian cools as droplets blasted into the air by a violent eruption; this is bollocks.