An Equip Card is a card that is equipped to a face-up monster on the field. Spell Cards with the Equip Icons 
are the most common type of Equip Card, and are officially called Equip Spell Cards. They are treated as both Spell Cards (they would be destroyed by the effect of "Heavy Storm") and Equip Cards (for purposes such as "Gearfried the Swordmaster".) Other cards may become Equip Cards through card effects, like with "Sword Hunter" or Union Monsters. Whenever a Monster Card becomes an Equip Card, it is treated as an Equip Spell Card, and is no longer considered a Monster Card. So a "Dark Magician" equipped to your "Sword Hunter" as an Equip Card could be destroyed by "Heavy Storm", but not by "Tribute to the Doomed".
Equip Cards constantly target the monster that it is equipped to, and they can subsequently destroy monsters that state they are destroyed if targeted by an effect. If an Equip Card's effect were to be negated (such as with cards like "Spell Canceller" or "Royal Decree" where applicable to the example), the card would remain "attached" to that same monster, and would still be subsequently removed from the field if the equipped monster were to be removed from the field or flipped face-down, because of Game Mechanics.
However, monsters equipped as Equip Cards still count as "being on the field", so effects or conditions that rely on a card "being sent from the field to the Graveyard", such as "Felgrand Dragon", would still apply. Despite this, they cannot activate/apply their effects on the field while treated as Equip Cards, as they are not "monsters" at the time.
Equip Cards whose equipped monsters have been removed from the field are destroyed and sent to the Graveyard afterwards, as they no longer have a target. In the case of many "Dragunity" and "Inzektor" cards, monsters are commonly equipped to other monsters as part of their effects, enhancing their usefulness. However, cards such as "Dragunity Aklys" need to be sent to the Graveyard while equipped for their effects to activate.
Equip Cards usually have beneficial effects applied to the equipped monster. Equip Cards can also be equipped to opponents' monsters to abuse the effect in some way, such as equipping your enemy's "Dark Magician" with "Big Bang Shot" and Setting a monster with low DEF. When the "Dark Magician" attacks, your opponent suffers the battle damage due to the wording on "Big Bang Shot". Note that while the monster who receives the benefit can be your opponent's, you are still the owner of the Equip Card, and thus any references to "you" still means the owner of the Equip Card itself, not the owner of the equipped monster.
Some Trap Cards, like "Blast with Chain", can also become Equip Cards. These cards are considered Equip Cards but are NOT Equip Spell Cards. This means that it is a Trap Card (and could be destroyed by "Remove Trap") and an Equip Card (for purposes such as "Gearfried the Swordmaster"). It is not a Spell Card, and cannot be destroyed by "De-Spell". We can call this kind of Trap an "Equip Trap Card" in a greater sense, although this is not an official designation.
For a list of Equip Spell Cards, see List of Equip Spell Cards.
For a list of non-Equip Spell Cards, see the list of cards Treated as Equip Cards.
List of cards that support Equip Cards: