What is a co-prime?

Two numbers are coprime if their highest common factor (or greatest common divisor) is $1$.

You can have the set of positive integers which are coprime to a given number: for example those coprime to $12$ are $1, 5, 7, 11, 13,17,19,23,25, $ and so on.

A number $a$ is not "a coprime"; rather, "coprimeness" is a relation that may or may not hold between two numbers $a$ and $b$. In other words, $a$ and $b$ may or not be "coprime to each other".

What does it mean for $a$ and $b$ to be coprime? That they share no common factors other than $1$, or equivalently, $\gcd(a,b)=1$ (where $\gcd$ denotes the greatest common divisor, see here).

For example, $2$ and $5$ are coprime, because if $d$ is a factor of $2$ and $d$ is also a factor of $5$ (i.e., $d$ is a common factor of $2$ and $5$), then $d$ has to be $1$ (or $-1$, technically). In other words, $\gcd(2,5)=1$.

However, $2$ and $6$ are not coprime, because they share the common factor $2$; and $\gcd(2,6)=2$.

As Qiaochu says, we can extend the definition of coprimeness to any collection of two or more numbers by declaring the set of numbers $\{a_1,a_2,\ldots\}$ to be coprime when any $a_i$ and $a_j$ are coprime (for $i\neq j$).

For more info see the Wikipedia page.

By definition, a pair of integers $\rm\:a,b\:$ are coprime if they have only trivial common factors, i.e. $\rm\:gcd(a,b) = 1\:,\:$ i.e. $\rm\:c\ |\ a,b\ \Rightarrow\ c\:|\:1\:.\:$ A set of integers is pairwise coprime if every pair from the set is coprime. The same definition works over any integral domain.

Coprime sets of integers share many of the properties of sets primes, e.g. factorizations into coprimes are unique. In fact in many number theoretic problems it suffices (and is more efficient) to work with coprimes rather than primes, e.g. see section $4.8$ on the concept of a gcd-free basis in Bach and Shallit: Algorithmic Number Theory.

Note: For ideals, some authors use coprime as a synonym for comaximal, i.e. $\rm\:I + J = 1\:.\:$