"Greater India" refers to the vast spread of Indian cultural (especially political and religious) formations throughout Southeast Asia that commenced in the early centuries CE and wasn't significantly curtailed until the arrival of Europeans beginning in the 15th century CE. This "Indianization" of Southeast Asia involved especially the spread of Indic language scripts and the popularization of ideas about political legitimization that combined aspects of vedic ritualism with Buddhist constructions of state organization and administration (otherwise known as rajadharma). We look at architecture especially in Southeast Asia as confirmation of this, as the plans of some of the grandest structures (think Angkor Wat, e.g.) are essentially monuments to Indian cosmologies and their attendant conceptions of sovereignty and economy.

Given this, it may be inaccurate to suggest that India didn't 'colonize' or 'conquer' part of the world. We can't say that India merely traded with Southeast Asia either, because Southeast Asian culture is so thoroughly Indianized.

Southeast Asia also arguably represents the most logical avenue for Indian expansion. There were formidable geographic barriers to he north and northwest, a thoroughly established China further to the east, and of course the juggernauts of Muslim and European societies to the far west.

Like many questions that get posed to professional historians, this one prefigures later events as the goal of the past. It's a common tendency in amateur -- and, truth be told, professional -- framing of the past. So it is important to remind ourselves frequently, to paraphrase E. P. Thompson, that the role of the historian is to save the past from the incredible condescension of the present. With questions like this, it's usually best to turn the microscope around and ask why European colonized when all of the so-called gunpowder empires -- the Ottoman, the Safavid, the Mughal -- as well as the Ming Dynasty possessed the technology and capacity for colonialism.

Though it is similar to other forms of conquest that span the classical and medieval ages, colonialism as practiced by Europeans beginning in the early modern period is unique in the way it married commerce and conquest. In Asia, colonialism was a way for the Europe's way muscle into a trade that they had encountered previously only at a great mark-up in the Mediterranean. What Europeans eventually encountered in Asia was a world of open trade that would have made Adam Smith blush.

At the risk of being overly simplistic, we can describe the great empires of the east as classic tribute empires whose wealth came from the taxation of agricultural production, either directly or through local nawabs, satraps, principalities, and landlords. For them, conquest meant taxation. The wealth from taxation was maximized by the ready availability of trade goods. The more trade, the more valuable their wealth. Therefore, the use of power to stifle and control the flow of commodities was anathema.

For Europeans, it was precisely the control of land and trade routes that offered them a place in this world. Before the flow of silver, gold, and later sugar from the Atlantic conquests, the Portuguese, Dutch, and British had few goods to offer this world. Their hopes for relevance grew with their capacity to control markets and industries they did not possess in Europe.

The Dutch in Taiwan, for example, invited Chinese craftspeople to make those products the Dutch had no hopes of producing themselves. Jumpcut to southeast Asia, the port city Melacca, prior to colonization, was defined by polyglot population and the free movement of traders. It was where east Asia met south Asia. First the Portuguese, later the Dutch, British, and Japanese controlled the city in turn. Melacca became stronghold for projecting control of land and especially trade. Not surprisingly, European control wasn't immediate and other cities took up the role Melacca played. Yet the contrast is important. Asian powers didn't colonize because it generally did not suit their interests. Europeans had no other entry into these markets.

As we know, aided by the unprecedented windfall of the Atlantic economy and the eventual industrial revolution, Europeans parlayed these outposts into the vast empires of a new sort, empires based on integrated global resources into what became by the end of the twentieth century a single global economy.

If you are interested in exploring the patterns of Asian trade further, I'd suggest Stewart Gordon's When Asia Was the World, which I sometimes assign in my world history class. Historiographic quibbles notwithstanding, it paints a compelling portrait into the how Asia operated prior to colonization, as well as explaining how European practices and strategies played when they first arrived.

I hope this helps and is not too didactic. Cheers!

Related question for experts: the medieval Chola dynasty was involved in overseas trade, had areas of influence in Indonesia and Malaysia along trade routes, and had a powerful navy. Can any similarities be drawn between the Chola combined martial/trade hegemony and "colonialism", or is such a comparison wholly anachronistic? Did Tamil influence in Indonesia have any of the hallmarks of "colonialism"?

Follow up question: From my limited understanding of Hinduism, voyaging by sea takes away your caste. Did this play a factor in the limited conquests of the Indian people?