Tedashi and tsumogiri

Tedashi and tsumogiri are terms that describe where in the hand a tile was discarded:

Tedashi means to discard a tile that was already in the hand before the draw, i.e. any tile other than the just-drawn tile.
Tsumogiri means to discard the just-drawn tile.

Reading opponent's tedashi and tsumogiri discards can be useful, as they can indicate what shapes are likely/unlikely to be in their hand. They are typically combined with other discard reading strategies.

When playing with physical tiles, players will have to manually check for discards. In online clients, it will vary. Notably, tenhou.net and Mahjong Soul have animations to display tedashi/tsumogiri when a tile is discarded, but do not indicate anything after the discard (forcing players to memorize the status of each desired tile).

Strategy

Learning to discard read

Beginners do not need to worry about discard reading - other things like tile efficiency are far more important at this stage.
- Being able to build hands efficiently is required for effective discard reading; discard reading requires knowing how players normally build their hands.
- Other beginners tend to not build hands in a consistent way, lowering the effectiveness of discard reading.
It is not required to memorize every single tile (e.g., if a guest wind is discarded turn 2, it doesn't matter if it's tedashi or tsumogiri). Try to focus on important tiles, such as dora and middle tiles.

Discard reading

Late tedashi tends to indicate that an opponent has other tiles nearby, but doesn't necessarily mean they will wait near said tile. E.g., a tedashi 3m might indicate they have a 34m ryanmen wait left, or a 345m complete sequence with a completely unrelated wait.
- Discarding a strong tile first, then a tedashi on a weak tile, such as tsumogiri 5 -> tedashi 8, also tends to indicate shapes nearby the weaker tile.
If an opponent tedashis a safe honor relatively late in the game, and there's no threat on the board, the opponent is likely in tenpai, or at least iishanten (where the honor was kept due to sakigiri).
If an opponent tedashis a joint/pair (e.g. tedashi 3p 5p), they will likely wait on a better joint. Tedashi ryanmen usually means they have a ryanmen-or-better wait.

Karagiri

Karagiri is when a player tedashis another copy of a tile that they just drawn (e.g., have 3-sou in hand, draw a 3-sou, then discard the 3-sou in hand rather than the just drawn 3).

Since karagiri gives out information about the hand, it should usually be avoided. Exceptions exist, though. The most obvious is to discard a 5 to accept a red 5. If it's all last and 1st place intentionally wants to deal in to end the game, you can karagiri to make your wait more obvious.