Definitions
from Wiktionary, Creative Commons Attribution/ShareAlike License
 adv. Too many to be counted (either by reason of being infinite or for practical constraints).
 adv. In an uncountable fashion.
 adv. In a way that is incapable of being put into onetoone correspondence with the natural numbers or any subset thereof.
 adv. Used as a general intensifier of amounts and quantities; very; much
Etymologies
from Wiktionary, Creative Commons Attribution/ShareAlike License
Examples

These sets are too big to be put into onetoone correspondence with the natural numbers; they are called uncountably infinite.

Thus, the onetoone correspondence between the reals and the naturals fails, as there are simply too many reals—they are "uncountably" numerous—making real infinity somehow larger than natural infinity.
Strange but True: Infinity Comes in Different Sizes  Impact Lab

For example, it certainly depends on whether your set of trials is countably infinite or uncountably infinite (in other words the cardinality of your set of trials).

* Eleanor Clift loses one of her uncountably many demerits in agitating for Howard Dean to HHS.

Much to the consternation of marine life advocates and to the relief of the outofsight/outofmind crowd, most of the damage is uncountably ensconced beneath the surface of the Gulf.

The questions raised by the uncountably infinite number of both unrealistic and implausible alternate viewpoints of diversity demand answers.

Or moving the other way, if we form a third language L³ by adding to L the quantifier Qx with the meaning “There are uncountably many elements x such that ¦”, then trivially L is reducible to L³, but the downward LoewenheimSkolem theorem shows at once that L³ is not reducible to L.

Or how might we have a little private time to tell just one of our sons of our affection for him without sharing the moment with uncountably many of his brothers?

Indeed, because cardinality is permutationinvariant, every cardinality quantifier is included, including “there are infinitely many”, “there are uncountably many”, and others that are not firstorder definable.

How can a countable model satisfy the firstorder sentence which “says that” there are uncountably many things?
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