"Nerf" is a term used to describe the altering of a card's effect or ruling, or of its Forbidden/Limited status, to make that card or a certain deck that revolves around that card far less powerful or more situational than it was previously. For instance, "Hyper Synchron" in the anime was given the effect of allowing a Synchro Monster to gain 800 ATK and the immunity from being destroyed by battle when it was used as a Synchro Material Monster. When it was printed as a real-life card in "Crimson Crisis", that effect was changed to giving an 800 point attack boost to a Dragon-Type Synchro Monster when it was used as a Synchro Material Monster, although the Synchro Monster would be Banished during the End Phase. Since the card was so drastically changed in effect to make it far less powerful and therefore less desirable, a player may see fit to call it "nerfed".
Another example is "Major Riot"; originally, its effect would activate whenever a monster was returned to its controller's hand, regardless of how it was returned. Recent rulings, however, changed it so that the monster must be returned by the opponent's effect before "Major Riot" can activate. Since the change in ruling severely reduced the amount of combos that could be used with it, it can be considered "nerfed".
A third example is "Berserk Dragon"; originally the card said "..can only be Special Summoned by the effect of "A Deal with Dark Ruler".", making it a Semi-Nomi monster when first released. It was later changed to "...except by...", changing the card to a Nomi monster. "Chaos Sorcerer" also says "...can only be Special Summoned by...", however, this was never changed to "Nomi". Both of the cards had the same text for Special Summoning except one was changed to Nomi.
A fourth example is "Earthbound Immortal Wiraqocha Rasca". In the show it has 1 ATK and DEF points and you can skip your Battle Phase to make your opponent's life points 1. When it became a real card it had 100 ATK and DEF points and when it is Normal Summoned you can return up to 3 cards you control to your Deck, except for Rasca, and discard the same number of cards from your opponent's hand at random. It would also gain 1000 ATK for each card discarded by this effect.
A fifth example would "The Winged Dragon of Ra". In the anime it had many special abilities, most notably when it was Special Summoned it was able to destroy and remove from play every monster the opponent had on the field and then attack the opponent directly; all of this while not being affected by card effects. When it was made into a legal card its ability to be Special Summoned was taken away altogether, as well as its immunity to card effects and some other abilities that made Ra practically unstoppable in the anime. It also became extremely weak - its base attack and defense were 0 instead of having an attack and defense equal to the sum of that of its tributed monsters.
A final example would be a change to the Forbidden/Limited List to limit "Thousand-Eyes Restrict", "Metamorphosis", and "Scapegoat" at the same time to curb the use of Goat Control Decks. Many players did not see the reason for limiting all three cards and this was further compounded when, in a later List, "Thousand-Eyes Restrict" was switched from Limited to Forbidden without altering the status of the other two cards. Therefore, the very Deck that these limits were supposed to curb was considered "nerfed" as the one major strategy of the Deck was greatly weakened and later nullified completely, undermining the entire point to the Deck. Even in Traditional Format, this Deck is still greatly weakened since there are very few cards to search for Quick-Play Spells.