How to skip over an element in .map()?

Just .filter() it first:

var sources = images.filter(function(img) {
  if (img.src.split('.').pop() === "json") {
    return false; // skip
  }
  return true;
}).map(function(img) { return img.src; });

If you don't want to do that, which is not unreasonable since it has some cost, you can use the more general .reduce(). You can generally express .map() in terms of .reduce:

someArray.map(function(element) {
  return transform(element);
});

can be written as

someArray.reduce(function(result, element) {
  result.push(transform(element));
  return result;
}, []);

So if you need to skip elements, you can do that easily with .reduce():

var sources = images.reduce(function(result, img) {
  if (img.src.split('.').pop() !== "json") {
    result.push(img.src);
  }
  return result;
}, []);

In that version, the code in the .filter() from the first sample is part of the .reduce() callback. The image source is only pushed onto the result array in the case where the filter operation would have kept it.

update — This question gets a lot of attention, and I'd like to add the following clarifying remark. The purpose of .map(), as a concept, is to do exactly what "map" means: transform a list of values into another list of values according to certain rules. Just as a paper map of some country would seem weird if a couple of cities were completely missing, a mapping from one list to another only really makes sense when there's a 1 to 1 set of result values.

I'm not saying that it doesn't make sense to create a new list from an old list with some values excluded. I'm just trying to make clear that .map() has a single simple intention, which is to create a new array of the same length as an old array, only with values formed by a transformation of the old values.

Since 2019, Array.prototype.flatMap is a good option.

images.flatMap(({src}) => src.endsWith('.json') ? [] : src);

From MDN:

flatMap can be used as a way to add and remove items (modify the number of items) during a map. In other words, it allows you to map many items to many items (by handling each input item separately), rather than always one-to-one. In this sense, it works like the opposite of filter. Simply return a 1-element array to keep the item, a multiple-element array to add items, or a 0-element array to remove the item.