Not to be confused with the Offset rule, a Counter is a chain that is capable of absorbing several lines of garbage and still be triggerable. Remember, no matter how much garbage the opponent sends you, the max that can fall at once is 5 lines (one rock). So ideally, you want your trigger to be at least 6 Columns higher than the next column over.
Holy Counter (ホーリーカウンター) refers to making a counter using only the left 2 Columns of the board. Any arrangement of Puyo that achieves that is considered a Holy Counter. But for reference, the most popular Holy Counter is the following form:
Evil Counter (イービルカウンター) refers to making a counter using the right 3 Columns of the board. Since you have 3 Columns to work with, it's easier to make the counter itself, but you'd be hard-pressed to make a decent tail.
Instead of 5 lines of garbage, Double Counters (ダブルカウンター) are designed to absorb 10 lines of garbage (two rocks). Thus, you have Double Holy Counters (ダブルホーリーカウンター) and Double Evil Counters (ダブルイービルカウンター).
Dam Counters (ダムンター) will power at the end.
The Matsukane Counter is built in the middle of the board instead of up against the wall. Can be useful as an "afterchain".
In a Successive Counter (サクセションカウンター), first you pop a chain on one side of the board, then you use the cleared space to absorb another rock and pop the chain on the other side.
Tanakacchi Style Counter
The Tanakacchi Style Counter (タナカッチ式カウンター) is another counter that's built in the middle of your board.
Ino Style Counter
The Ino Style Counter (イノ式カウンター) can counter on the left, or you can pop the groups on the right for a Power Chain. There's also less counting involved in Column 3.