I'm a topologist, and so this answer is going to specifically be about analogies with ordinary topology. I like to think of different Grothendieck topologies as corresponding to different "allowable" ways to build up a space using a quotient topology.

1. The Zariski topology is like recognizing that a space $X$ can be built up from a collection of open subsets $U \subset X$ which cover $X$ and identifying points in common intersections.

2. The etale topology is like recognizing that if a map $Y \to X$ is surjective and a local homeomorphism, then this map makes $X$ homeomorphic to the quotient of $Y$ by an equivalence relation.

3. The fppf topology is similar to the etale topology, but now allowing maps $Y \to X$ which are open.

4. The fpqc topology, for lack of a better analogy, is like allowing arbitrary maps $Y \to X$ which make $X$ into a quotient space of $Y$.

Much of the study of these topologies is the study of sheaves on them. The choice of topology has at several significant consequences.

1. The choice of topology determines how much functoriality your sheaf has. (In particular, choices like "big site" or "small site" determine how many objects $Y$ you can evaluate your sheaf on!)

2. The choice of topology determines what kind of data you need to construct things (which is code for descent theory). We know that if $Y \to X$ is a quotient map, we can build a vector bundle on $X$ by taking a vector bundle on $Y$ and imposing a compatible equivalence relation on it (e.g. from a clutching function). Similarly, to build a sheaf in topology $\tau$ we just need to build it on a $\tau$-cover and glue it together.

3. The choice of topology determines your definition of "local". I really like Simon Pepin Lehalleur's comment here, because it describes exactly the following problem: when is a map ${\cal F} \to {\cal G}$ of sheaves surjective? It is surjective when it is surjective _locally_. For example, the n'th power map $\Bbb G_m \to \Bbb G_m$ is a surjection of sheaves precisely when, _locally_, every invertible function is an n'th power of some other invertible function. That's rarely true in the Zariski topology; it's true in the etale topology when $n$ is invertible, because then adjoining a solution of the polynomial $(x^n - \alpha)$ is an etale extension; it's always true in the fppf topology because adjoining a solution of the polynomial $(x^n - \alpha)$ is always fppf.