Why did European colonization happen?

There is no single answer as to why European colonization
occurred; as each country had it's own motives. Originally, the primary aim of
exploration was to find an alternative route by sea to the riches of the East. Columbus
himself was attempting to do just that. Once the New World was discovered, there was
little interest in colonizing it for some time.

The first
to exploit the New World were the Spanish who were primarily interested in gold and
silver. Their secondary interest was in converting the Indians to Christianity, but they
did not let this stand in the way of their primary goal. The Spanish established a
settlement at St. Augustine, Florida, the earliest permanent settlement in the U.S. as
an outpost to guard their gold shipments to Europe.

English
colonization was at first an effort by Sir Walter Raleigh and his half brother, Sir
Humphrey Gilbert, to establish an outpost to keep an eye on Spanish settlements. They,
of course, failed miserably, notably the "lost colony" of Roanoke. English colonization
was justified by Sir Richard Hakluyt in his Discourse on Western
Planting:

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That this westerne discoverie will be greately
for the inlargement of the gospell of Christe whereunto the Princes of the refourmed
relligion are chefely bounde amongest whome her Majestie is principall. That all other
englishe Trades are growen beggerly or daungerous, especially in all the kinge of Spaine
his Domynions, where our men are dryven to flinge their Bibles and prayer Bokes into the
sea, and to forsweare and renownce their relligion and conscience and consequently theyr
obedience to her Majestie.That this westerne voyadge will yelde unto us all the
commodities of Europe, Affrica, and Asia, as far as wee were wonte to travell, and
supply the wantes of all our decayed trades. That this enterprise will be for the
manifolde imploymente of nombers of idle men, and for bredinge of many sufficient, and
for utterance of the greate quantitie of the commodities of our Realme.That this voyage
will be a great bridle to the Indies of the kinge of Spaine and a means that wee may
arreste at our pleasure for the space of teime weekes or three monethes every yere, one
or twoo hundred saile of his subjectes shippes at the fysshinge in Newfounde Iande. That
the rischesse that the Indian Threasure wrought in time of Charles the late Emperor
father to the Spanishe kinge, is to be had in consideracion of the Q. moste excellent
Majestie, leaste the contynuall commynge of the like threasure from thence to his sonne,
worke the unrecoverable annoye of this Realme, whereof already wee have had very
dangerous
experience.

Notwithstanding
Hakluyt's high sounding principles, the colony at Jamestown was founded by a joint stock
company with the intent of profiting financially. Northern colonies, such as Chesapeake
Bay were intended to be representations of model Christian communities, a "City on a
Hill" to use Winthrop's phrase.

Finally the French came for
economic motives and to convert the Indians; their motivation being the fur trade.
Ultimately, only the English were successful.