Eggs are a nickname given to minions with similar abilities: they feature 0 Attack, and a Deathrattle effect far above their mana cost. Most also have 'egg' in their name, although not all egg-like cards are eggs and not all cards that are eggs are "egg" effects.
Eggs are great cards for Zoo decks, which want to control the board. Eggs contribute to this by providing a benefit above their original mana cost once broken. On a baseline, they are a great way to deter your opponent from using board control against you. However, they become most useful when the minion's original stats are made use of, such as by cards like Abusive Sergeant, Power Overwhelming, or Grim Rally.
Despite their incredible power in the right circumstances, eggs are some of the worst top decks an aggressive deck can have. Because of this, they are traditionally most commonly used in Warlock thanks to both Warlock's strong synergy and Life Tap making up for the inherent inconsistency of the cards, or Deathrattle Hunter in Wild.