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Sifter is a broad term that applies to two types of cards:

Deck inspectors that allow you to discard cards from the top of your deck
Cards that draw and force you to discard, or allow you to discard in order to draw

Both are useful for cycling through your deck to get to more desirable cards. Many of these are non-terminal Actions, such as Warehouse. Sifters can increase or decrease handsize, depending on what additional role they take, if any. Some draw-to-X effects are also sifters because they make you discard your hand before drawing.

Card that do something other than drawing cards as compensation for (usually optional) discarding are discard for benefit cards. Cards that reveal (and eventually discard) cards while looking for specific other cards are digging cards.

List of sifters

Cards in italics have been removed.

Deck Sifters

Most of these are also deck inspectors.

Deck Sifters that are not Deck inspectors

These allow you to sift through your deck, but do not allow you to put cards back:

Catacombs
Border Guard
Miller
Hunter - may force discarding cards of certain types

Hand Sifters

These cards draw cards then discard, or vice versa.

Non-terminal

Cellar
Diplomat - sifts as its Reaction effect (using the Reaction does not use up an Action)
Secret Passage - puts a card back in your deck, rather than discarding
Warehouse
Oasis
Inn
Ferryman
Stables - Only lets you discard Treasure cards
Fugitive
Dungeon
Forum
Shepherd
Old Map
Innkeeper
Capital City
Grotto
Rustic Village
Alley

Terminal

Sea Witch
Young Witch
Embassy
Witch's Hut
Storeroom
Royal Blacksmith - Only discards Coppers, but discards all of them

Landscapes

Draw-to-X

Some draw-to-X effects are also sifters because they make you discard your hand before drawing.

Strategy

Sifters, particularly those that give +1 Action, are most useful in trying to get two combo-ing cards to collide. The most extreme case is Treasure Map, but there are other cards that do best in combination with another card. Sifters are optimal for this because they usually draw more than terminal draw cards at the same cost, giving you more cards that could be what you're looking for, and since you're only looking for two cards, you don't care that you have to discard afterwards. All sifters are also quite nice with Tunnel, as they provide the much-needed discard outside of the Clean-up phase to gain Tunnel's Golds.

Sifters which maintain or increase handsize and sifters which draw-then-discard are much more flexible than those which discard first or shrink the player's hand. For example, a single copy of Cellar can provide great value to an engine, but a hand with two cellars forces an awkward choice. On the other hand drawing two copies of Forum is almost strictly better than drawing one, since they require no guesswork and can be chained in any number. Discard-first and hand-shrinking sifters can add consistency to an engine but rarely enable one, while cards like Forum can enable engines even in the absence of Village or Library effects.

Terminal sifters are usually more useful in the context of an engine, or in big money strategies. Without village support, they are highly likely to draw Actions dead, given their higher draw, though this is mitigated by their discarding requirement. This makes them very useful for big money, as they can concentrate your Treasures in hand while skipping over Victory cards. Terminal sifters are especially improved by on-demand +Actions like Villagers and Coin of the Realm, since these effects allow you to sift for villages rather than having them in your starting hand.