Hand Extension is an informal game term. Extending your hand means to make more than eight cards at a time available during your Shadow Phase. This allows you to play more minions or other cards so you don't waste any Twilight, especially if your opponent floods. Hand extension is very important to certain Shadow strategies, especially swarm strategies that want to play as many minions as possible in a single turn. Most Shadow cards are disposable and discarded at the end of the turn, so playing as many as you can as quickly as you can is often important.
Usually, hand extension means drawing extra cards at the start of or during the Shadow Phase, using cards like Bent on Discovery (1R206) or Grishnákh, Orc Captain (5R100), or reconciling early, with a card like Under Foot (10C52). However, that isn't the only way to extend your hand. Sometimes you can play cards stacked on other cards: Goblin Swarms (1R183) can store minions for re-use, or Dunlending Looter (4U11) can stack himself on a site you control then replay himself in a later turn. Sometimes you can play minions directly from your discard pile, with cards like Uruk Captain (2R46) or Gollum, Plotting Deceiver (7R58). Sometimes you can even recur cards from your discard pile straight to your hand, with cards like The Witch-king, Morgul King (7R221) or Goblin Hordes (11R123). As long as you're playing more cards that you would've been able to draw normally by reconciling at the end of the last regroup phase, that's hand extension.
Generally, drawing cards after the end of the Shadow Phase isn't considered hand extension unless you can also somehow get extra cards into play by doing so. A Shadow Fell Over Them (15R182) can extend your hand (albeit at a high Twilight cost). However, using Plundered Armories (1C193) to bounce a Goblin Scimitar (1C180) from orc to orc as they die in the Skirmish Phase lets you draw extra cards, but you probably can't play them this turn. Likewise, Shadow cards played in the Regroup Phase don't help with the skirmish, so a card like Gothmog, Morgul Leader (18R80) is not a hand extender.
Free Peoples decks don't usually need to play all their cards at once quite as badly as Shadow decks do. As a result, hand extension isn't a term that applies to Free Peoples decks: instead, they are usually designed with an eye to cycling and filtering out your deck.