Cycling is a keyword ability.
Typecyling is a variant on cycling, where rather than drawing a card, you instead may search your library for a card of the specified type.
Amy has a Lonely Sandbar in her hand and wishes to cycle it. To do so she must pay and discard the Lonely Sandbar from her hand. When the ability resolves, she will then draw a card.
Alternatively, Arnold has an Absorb Vis in his hand and wishes to cycle that. Similarly, he must pay and discard the Absorb Vis. However, this time, when the ability resolves, he will instead search his library for a Basic Land card, and put it into his hand.
- 702.28a. Cycling is an activated ability that functions only while the card with cycling is in a player's hand. "Cycling [cost]" means "[Cost], Discard this card: Draw a card."
- 702.28b. Although the cycling ability can be activated only if the card is in a player's hand, it continues to exist while the object is on the battlefield and in all other zones. Therefore, objects with cycling will be affected by effects that depend on objects having one or more activated abilities.
- 702.28c. Some cards with cycling have abilities that trigger when they're cycled. "When you cycle [this card]" means "When you discard [this card] to pay a cycling cost." These abilities trigger from whatever zone the card winds up in after it's cycled.
- 702.28d. Typecycling is a variant of the cycling ability. "[Type]cycling [cost]" means "[Cost], Discard this card: Search your library for a [type] card, reveal it, and put it into your hand. Then shuffle your library." This type is usually a subtype (as in "mountaincycling") but can be any card type, subtype, supertype, or combination thereof (as in "basic landcycling").
- 702.28e. Typecycling abilities are cycling abilities, and typecycling costs are cycling costs. Any cards that trigger when a player cycles a card will trigger when a card is discarded to pay a typecycling cost. Any effect that stops players from cycling cards will stop players from activating cards' typecycling abilities. Any effect that increases or reduces a cycling cost will increase or reduce a typecycling cost.