That's a rather difficult question - when you look at how European influence developed and expanded, it gets quite hard to pin it down to a few factors.
There are also different kinds of dominance - for example, the nature of the Spanish Empire was very different to the nature of the later British.
Europeans first started having influence in the east with the rise of chartered trading companies - who could go overseas and be legally guaranteed a monopoly of whatever products they could bring back. Obviously to continue trading they'd need bases of some sort in the area and also the means to defend themselves against pirates and other such threats.
European expansion really kicked off when companies were competing or if they received trouble from the natives. For an example of the latter, in 1756, the new ruler of Bengal seized the British East India Company's settlement of Calcutta - Clive retaliated and then pushed forward and defeated the Indians at Plassey next year.
An example of the former would be Wellseley in India in the early 19th/late18th century: It was well known that Napoleon wished to threaten British commerce in India, so the Company exerted further dominance in the region to counter this.
There were other ways Europeans ended up expanding - for example, if the company got along peacefully, the area would just be outright colonised (like the American colonies).
[to be continued in another post]