Transparency is given by an alpha channel. It can be a different image. Black transparent, white opaque.
Brightness on the image itself is indicated by a luma channel. Black, the darkest point. White is the brightest.
Not every file format stores the information using the same channels. They can be transformed or extracted later.
In a digital image, one channel is one set of numbers or values with some information.
Not every file format stores the image information the same way or in the same type of channels. But when the image is open in an image editor, the information can now be implied and then extracted.
What do I mean by implied?
Let's say we have one pixel whose color is saved as RGB with the values R200 G200 B200.
I can imply that that pixel has no saturation, because all channels have the same value, and it is somehow bright.
Now we have one pixel whose value is R50 G0 B0. I can say that it is a dark color, but has some good saturation because the difference between the red and the other two is significant.
Now, the answer to your question.
An alpha channel stores information that usually is used to produce transparency (or opacity which is the opposite word). (I say usually because, in some file formats, you can simply store several alpha channels, and then use them later, for different things)
This channel could be a totally different image by itself. In this example, the alpha channel is the image with some waves on it.
Black is full transparency (Or 0% opacity). White is 100% opacity. And any gray value is semi-transparent accordingly.
Channels that actually compose an image
Color can be mapped in different 3-axis forming one 3D solid (Like a chromatic circle but in 3D). This can be different models, for example, RGB.
You can clearly see that a bright zone in the red channel (and black on the other two) indicates a lot of red in the image (R).
But when compression is important, for example in JPG images or video, you usually use some variation of a model based on Luminance, this is brightness.
This is because our eyes distinguish a lot more differences in brightness than other information. And the color information itself is stored in some other two channels.
Here is an actual separation of a luma channel, also called the Y channel, and two chroma channels Cr and Cb (Red and Blue components)
You can see how the luma is the one that carries the most information for a recognizable image. The other two are not that detailed (Therefore easier to compress)
The color information on the file itself is not stored in RGB channels. But when opened you can now play with those RGB values.
Some other separation of channels using brightness
Here I separated the same image in HSB.
You can see that now, the bright zone on the Saturation channel tells you that it has a very saturated color (S).
But the main information about the image is on the brightness channel (B).
The grayscale image looks brighter than (Y) because of the differences in the scales between the models.