Fifteen years of vision architecture research boiled down to a handful of ideas that survived contact with reality: go deeper, but only once you can actually train the depth; use small filters repeatedly rather than large ones once; and share computation aggressively when compute or memory is scarce. Nearly every architecture below is one of these ideas, applied and refined.
A CNN's spatial resolution has to shrink somewhere between a 224×224 input and a single classification decision — pooling is the classic way to do that shrinking, trading spatial precision for a degree of invariance and reduced compute. Getting the resulting shapes right, at every layer, is the single most useful bookkeeping habit for building a CNN that actually runs.
A fully-connected layer applied directly to a 224×224 RGB image would need over 150,000 input weights per single output unit — and that's before considering how many units a layer needs. Convolution replaced that with a small, shared filter slid across the image, cutting parameters by orders of magnitude while adding a property fully-connected layers structurally lack: the same filter finds the same pattern no matter where in the image it appears.