Bhante is a contracted vocative form of the reverential "bhadanta", which is simply an appellative used to show respect. It is used in the suttas in a similar way as the word "Ajān" is used in Thailand. Strictly speaking, it should not be used when referring to a respected individual in the third person, e.g. "Bhante Bodhi", etc., and it isn't generally used to address someone of equal stature ("bho" is commonly used in that case).
It really is the most common form of address used by lay people to address both the Buddha and his monastic disciples. As was quoted in another answer, after the Buddha passed away, it also became the method for junior monks to address senior monks (individually or as the sangha).
There is also an instance in the Vimanavatthu where a lesser angel uses Bhante to address the king of the angels.
For the Buddha (addressed by King Ajatasattu):
tassa me, bhante bhagavā accayaṃ accayato paṭiggaṇhātu āyatiṃ saṃvarāya
"Thus may you, Bhante Bhagavā, accept my fault for a fault, for the purpose of guarding from this point on."
-- DN 2
For a respectable monk (Nagita, addressed by a Licchavi householder):
kahaṃ nu kho, bhante nāgita, etarahi so bhagavā viharati arahaṃ sammāsambuddho
"Where exactly, Bhante Nāgita, at this time, is that Bhagavā dwelling, the worthy, fully-self-enlighted Buddha?"
-- DN 6
For a junior monk to address a senior monk (Ananda addressing Anuruddha):
parinibbuto, bhante anuruddha, bhagavāti. nāvuso ānanda, bhagavā parinibbuto, saññāvedayitanirodhaṃ samāpannoti.
"Is the Bhagavā completely extinguished, Bhante Anuruddha?" "The Bhagavā is not, Āvuso Ānanda, completely extinguished; [he is] attained to the cessation of perception and sensation."
-- DN 16
For an ordinary monk, just because he is a monk (Udāyi, addressed by the carpenter Pañcakanga):
na kho, bhante udāyi, tisso vedanā vuttā bhagavatā
"Not so, Bhante Udāyi, were three feelings spoken of by the Bhagavā."
-- MN 59
For the king of angels (addressed by a lesser angel):
kosātakī nāma latatthi bhante, tittikā anabhicchitā.
"There is a creeper called Kosātakī, Bhante, bitter and undesired."
-- Vv. 798