Buddha means the awakened one. As he fully comprehends the four noble truths, and arises from the allays of ignorance, he becomes a Buddha. Buddha is not a god, he is not an human, he is not some person who is born as one. Buddha is not born, a person becomes Bodhisattva first. And then with his own efforts, through spiritual ablution and development of basic virtues as, discipline, generosity, wisdom, endurance, truthfulness, selflessness, benevolence etc. a mortal attains Buddhahood. Enlightenment is a fully awakened state of vast wisdom through which reality in all its complexity can be fully understood and enjoyed. Any human being who is awakened to the fundamental truth about life can be called a Buddha. Buddhahood is a dynamic process, and achieving it is as arduous as it can be.

Siddhartha Gautama was born in 623 B.C. (according to UNESCO) to King Suddhodhana and Maya Devi, as a prince of Kapilvasthu, and ancient kingdom which lies in the Southern state of present day Nepal. Although there are doubts about birthplace of Siddhartha Gautama, especially as an Indian Propaganda, the Pillar of Asoka, and ancient inscriptions very much indicate he was in fact born in Lumbini, Nepal.

Siddhartha Gautama was a compassionate man who rejected the joys of being a prince, both asceticism and attachment, and set foot to the path for attainment of Buddhahood. Siddhartha Gautama attained Buddhahood at the age of 35, after 49 days of meditation (which was a middle way adopted by him after many efforts of attaining Buddhahood through finding guru, abstaining ascetics and abnegation failed), in a place called Bodhgaya under tree shade (Bodhi tree), presently in Uttar-Pradesh, India(?). Siddhartha Gautama, hence become Buddha, Gautama Buddha or Shakyamuni as many refer today.

After becoming Buddha, he first gave his sermon to 5 companions in Banarasi, thus forming first sangha; company of Buddhist monks. For the remaining 45 years of his live, he travelled throughout the Indian Subcontinent, perching and teaching people of all caste and origin, people of all profession, even the cannibals; what it is now called as four noble truth and noble eight fold path. His teachings and disciples formed a religion now called as Buddhism. (oh! And yes Gautama Buddha was an iconoclast, funny enough, the largest number of idols in the word today are his.)

So, where was Buddha born? He wasn't, in-fact it's not "him". There are many Buddha, anyone who attains Buddhahood is  a Buddha.

However, it is common practice that it is Shakyamuni or Siddhartha Gautama whom we regard as Buddha generally. And there is no denying the fact that Siddhartha Gautama was born in ancient Kapilvasthu kingdom (which lies in Present day Nepal).

Here is a google earth snapshot of famous Lumbini region including Maya Devi temple, Sacred ponds and other shrines (Birth place of Lord Buddha).