The Basics
Start solving the board from the top. Let's focus on one row at a time, and ignore what the rows below the one we're working on look like.

(The following images have Light tiles only in the first row, since we're focusing on the first row. That's not how your board is supposed to look at the beginning of a level, so don't worry if it doesn't.)

You can move a tile to the left by clicking the tile that is to the bottom left of it; you can move a tile to the right by clicking the tile that is to the bottom right of it.

If you have two tiles next to each other, clicking the tile under and between them will clear both.

If a tile is near the left edge, click the tile to the bottom left of it to clear it; similarly, if a tile is near the right edge, click the tile to the bottom right of it to clear it.

If you have a single tile that isn't near an edge, move it to the left or right until it is near an edge, then clear it.

If you have two tiles that aren't next to each other, you can either move them until they are next to each other then clear them...

...or you can clear one of them using the left edge and the other using the right edge.

Once the first row is clear, do the same with the second row, then the third row.

From the fourth row onwards, you can only clear tiles by pairing them up.

You can't use the edge to get rid of tiles, since in all rows below the third the first tile doesn't have a tile to the bottom left of it and the last tile doesn't have a tile to the bottom right of it.

On average, all of the following methods require the same amount of time/turns (Method 3 is a little less efficient than the other two).

I suggest trying them all out and finding the one that feels more natural to use and easier for you to memorise.

Method 1
Make the first row all one color by bringing down the tiles of the other color, then do the same with the second row, the third row, all the rows one by one.

If you're lucky, you might be able to proceed this way until you clear the whole board.

If that's not the case, you either won't be able to bring all your tiles to the bottom or you will have a number of leftover tiles in the bottom two rows.

If you weren't able to bring all tiles to the bottom because you have an odd number of tiles in the fourth row, click all tiles in the fourth row and continue solving.

If you have an odd number of tiles in the fifth row, click all tiles in the second row and bring the resulting tiles down (the most efficient way to do it is clicking the tiles shown below in the third row).

Tip: when you're bringing down the tiles from the second row and you reach the moment when you have to bring them into the fourth row, clear the tiles by pairing them off.

If you clear the first one by itself then the next two together and the last one by itself, you will get an odd number of tiles in the fourth row and you'll have to perform the fourth row solving click pattern, which will take more time.

If you have an odd number of tiles in the sixth row, click the top two corners and bring the resulting tiles to the bottom as shown in the following image.

(You don't need to memorise this whole click pattern; when you're solving a level, you will be able to figure out which tiles you need to click to work your way down after clicking the top corners. This just shows the fastest way to do so, in case you're unsure how to proceed.)

You will then be able to completely clear the board.

If you're solving correctly, you shouldn't need to use any of these click patterns more than once in the same level, thought you might need to use all three in a level.

Are you having trouble figuring out how to apply these solutions to an actual level?

This video (link leads to YouTube) by @/sneekiblin shows Method 1 in action.

Method 2
This method is originally from

this video (link leads to YouTube).

Clear the tiles of the undesired color from the board working from the top down and from left to right. That means that if you have lone tiles in the first three rows, you use only the right edge to get rid of them, not the left edge.

You might be able to completely clear the board this way, but more likely you will eventually have leftover tiles in the highlighted area. If you have leftovers outside that area, move them to the right until you only have leftovers in the highlighted area.

Let's consider only the leftover tiles in the bottom row, since these leftover configurations are always symmetrical.

If tile A is of the undesired color, click tile 1.

If tile B is of the undesired color, click tile 2.

If tiles C and D are different from each other, click tile 3.

Bring the resulting tiles down working again from left to right. You will then be able to completely clear the board.

Method 3
This method is a mix of the first two.

Clear the tiles of the undesired color from the board working from the top down and from left to right. That means that if you have lone tiles in the first three rows, you use only the right edge to get rid of them, not the left edge.

If you weren't able to bring all tiles to the bottom because you have an odd number of tiles in the fourth row, click the first tile in the first row and bring the resulting tiles down.

If you have an odd number of tiles in the fifth row, click the second tile in the first row and bring the resulting tiles down.

If you have an odd number of tiles in the sixth row, click the third tile in the first row and bring the resulting tiles down.

(You don't need to memorise these whole click patterns; when you're solving a level, you will be able to figure out which tiles you need to click to work your way down. The images just show the fastest way to do so, in case you're unsure how to proceed.)

You will then be able to completely clear the board.

If you're solving correctly, you shouldn't need to use any of these click patterns more than once in the same level, thought you might need to use all three in a level.