Trapusa and Bahalika (alternatively Bhallika) are attributed to be the first two lay disciples of the Buddha. The first account of Trapusa and Bahalika appears in the Vinaya section of the Tripiṭaka where they offer the Buddha his first meal after enlightenment, take refuge in the Dharma (while the Sangha was still not established), and become the Buddha's first disciples. Xuanzang says that Buddhism was brought to Central Asia by Trapusa and Bahalika (referring to Balkh) two merchants who offered food to the Buddha after his enlightenment.
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The era of Trapusa and Bahalika is during the life of the Historical Buddha: most early 20th-century historians dated his birth and death as c. 563 BCE to 483 BCE, but more recent research dates his death to between 486 and 483 BCE or, according to others, between 411 and 400 BCE.
Central Asia
Xuanzang recounts, having become his first disciples Trapusa and Bahalika wished his leave to return home, they asked the Buddha for something by which they could remember and honour him in his absence. The Buddha gave them eight of his hairs as relics. They made golden caskets for the relics and took them to their own city (Balkh) where they enshrined them in a stupa by the city gate. See Nava Vihara.
Xuanzang recounts that theirs was the first ever Buddhist Stupa to be made and that the Buddha had first to instruct them how to erect it by folding his three robes into squares piling them up and then topping them off with his inverted bowl.
Sri Lanka
What is stated in the chronicles of Sri Lanka is that, two merchant brothers, Thapassu and Bhalluka, who travel to different parts in the region, met the Buddha just after his Enlightenment while they were on their way to Rajagaha (Rajgir). They offered the first alms to the Buddha. Taking refuge in the Buddha and the Dhamma they became the first lay disciples of the Buddha. Requesting the Buddha something to worship, the Buddha gave them, eight handfuls of his hair, which he obtained by stroking his head. During their travel they arrived at Thiriyaya (North-East of Sri Lanka) from India. At Thiriyaya, some of the hair relics they brought were enshrined at a Stupa built on top of a mountain which is now called Girihandu Seya. That is considered the first Stupa in Sri Lanka.
Myanmar
According to tradition, Taphussa and Bhallika — two merchant brothers from the city of Balkh in what is currently Afghanistan — met the Lord Gautama Buddha during his lifetime and received eight of the Buddha's hairs. The brothers traveled to Myanmar and, with the help of the local ruler, King Okkalapa, found Singuttara Hill, where relics of other Buddhas preceding Gautama Buddha had been enshrined. This is now called Shwedagon Pagoda.
Significance
John S. Strong draws attention to Trapusa and Bahalika's legacy of pioneers:
We thus have an important tradition here that brings together several "firsts": first lay disciples of the Buddha to take refuge in him and his teachings; first meritorious food offering to the Buddha after his enlightenment; first Buddhist monk's bowl; first words of dharma given by the Blessed One; first relics of Gautama after his attainment of buddhahood; and first Stupa of the Buddha here on earth.