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An "Equip Card" is any card that is equipped to a face-up monster on the field. Spell Cards with the Equip Icon 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". However, monsters equipped as Equip Cards still count as "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 even for monsters treated as Equip Cards.
Equip cards usually have beneficial effects applied to the equipped monster. Equip cards can 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 set a monster with low def. When the dark magician attacks, your opponent suffers the lifepoint damage due to the wording on Big Bang Shot). Note that, while the monster who recieves 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".