The most abstract way of defining a group is to use a group presentation. This is a set of distinct identified group elements (known as generators) together with a set of relations between these generators. Every group has a presentation (but different presentations may define the same group).
For example, the cyclic group with $n$ elements is defined by taking a single generator $a$ and applying the relation $a^n=1$. This is notated as $\langle a | a^n=1 \rangle$ (sometimes the "$=1$" part in relations is taken as read and omitted). The dihedral group with $2n$ elements has two generators $a$ and $b$ and presentation $\langle a,b| a^n=1, b^2=1, (ab)^2=1 \rangle$.
To avoid ambiguity, we say that a presentation defines the largest possible group that satisfies the given relations - so the presentation of the cyclic group $C_2$ is $\langle a|a^2=1\rangle$, even though $C_2$ also satisfies $\langle a|a^{2m}=1\rangle$ for any integer $m$.
A group presentation defines the rules that govern composition of the group's elements in the most abstract way, and does not rely on any representation or realisation of the group. However, the relations can seem arbitrary, and it is often not the most "intuitive" way to define a group.