My lecturer mentioned today that it was possible to "label" loops in Java so that you could refer to them when dealing with nested loops. So I looked up the feature as I didn't know about it and many places where this feature was explained it was followed by a warning, discouraging nested loops.

I don't really understand why? Is it because it affects the readability of the code? Or is it something more "technical"? 

Nested loops are fine as long as they describe the correct algorithm.

Nested loops have performance considerations (see @Travis-Pesetto's answer), but sometimes it's exactly the correct algorithm, e.g. when you need to access every value in a matrix.

Labeling loops in Java allows to prematurely break out of several nested loops when other ways to do this would be cumbersome. E.g. some game might have a piece of code like this:

Player chosen_one = null;
...
outer: // this is a label
for (Player player : party.getPlayers()) {
  for (Cell cell : player.getVisibleMapCells()) {
    for (Item artefact : cell.getItemsOnTheFloor())
      if (artefact == HOLY_GRAIL) {
        chosen_one = player;
        break outer; // everyone stop looking, we found it
      }
  }
}


While code like the example above may sometimes be the optimal way to express a certain algorithm, it is usually better to break this code into smaller functions, and probably use return instead of break. So a break with a label is a faint code smell; pay extra attention when you see it.
