I've never really done much game programming before, pretty straightforward question.

Imagine I'm building a Tetris game, with the main loop looking something like this.

for every frame
    handle input
    if it's time to make the current block move down a row
        if we can move the block
            move the block
        else
            remove all complete rows
            move rows down so there are no gaps
            if we can spawn a new block
                spawn a new current block
            else
                game over

Everything in the game so far happens instantly - things are spawned instantly, rows are removed instantly etc. But what if I don't want things to happen instantly (i.e animate things)?

for every frame
    handle input
    if it's time to make the current block move down a row
        if we can move the block
            move the block
        else
            ?? animate complete rows disappearing (somehow, wait over multiple frames until the animation is done)
            ?? animate rows moving downwards (and again, wait over multiple frames)
            if we can spawn a new block
                spawn a new current block
            else
                game over

In my Pong clone this wasn't an issue, as every frame I was just moving the ball and checking for collisions.

How can I wrap my head around this issue? Surely most games involves some action that takes more than a frame, and other things halt until the action is done.