the word metric
Take a group and a chosen set of 'basic moves' (generators). Any element of the group is reached from the identity by a sequence of these moves, like a recipe spelled out as a word. The word metric measures the distance between two group elements as the least number of basic moves needed to get from one to the other. It is the device that turns an abstract algebraic group into a concrete metric space you can do geometry on.
Precisely, let G be a group with a finite generating set S, which we take symmetric (if s is in S so is s^{-1}). The word length |g|_S of an element g is the smallest n such that g can be written as a product s_1 s_2 ... s_n of generators from S. The word metric is then d_S(g, h) = |g^{-1} h|_S, the word length of the element that takes g to h. This is a genuine metric, and it is left-invariant: d_S(ag, ah) = d_S(g, h) for every a, so multiplying everything on the left by a fixed group element is an isometry — the group looks the same from every point. Computing a distance means solving a shortest-word problem: for example in the integers Z with generator S = {+1, -1}, the word length of n is just |n|, so the word metric on Z is the ordinary distance on the integer line.
The word metric is the entryway to geometric group theory: it lets you ask geometric questions about a group — is it hyperbolic, how fast does it grow, what does it look like from far away. The unavoidable caveat is the dependence on S. Different finite generating sets give different word metrics, and the actual distances change. But they change in a controlled way: any two word metrics from finite generating sets on the same group are bi-Lipschitz equivalent, hence quasi-isometric, so all the coarse, large-scale invariants (growth type, hyperbolicity, number of ends) are independent of the choice. You should never attach meaning to an exact word distance as a group invariant — only to the quasi-isometry class.
Take G = Z^2 with the standard generators S = {(1,0), (-1,0), (0,1), (0,-1)}. The word length of (a, b) is |a| + |b|, the taxicab (Manhattan) distance, because the cheapest way to reach (a,b) is |a| horizontal steps plus |b| vertical steps. So the word metric on Z^2 is the L^1 metric on the integer grid. If instead you added the diagonal (1,1) to S, distances would shrink (you could move diagonally in one step) — a different but quasi-isometric metric.
Standard generators make Z^2's word metric the taxicab distance |a| + |b|.
The word metric depends on the chosen generating set, so exact distances are not group invariants. Only quasi-isometry-invariant features (growth, hyperbolicity, ends) are intrinsic; change S and the numbers change, the coarse geometry does not.