The process of mixing picks up right where the production part left off. Traditionally, when you reached the mixing stage, you can not edit the tracks, add new tracks or alter the overall arrangement of the song. With the power of modern music production software this is no longer the case, but for the sake of simplicity we will stick to the restrains of the old world.
When you are mixing material that you recorded yourself you likely already have the finished mix in your head. You have the benefit of knowing how all the tracks were created, what challenges you could already foresee and what the artists intention was with how everything was put together. This can be of great value but also deprives the mixing engineer from the very important "first listen". Our perception of a song changes drastically the more often we hear it and under audio professionals this "first listen" perspective is the reason why many outsource the mixing and mastering stage. The longer you are "in the trenches" with a project, the higher the risk that you might loose sight of the big picture.
So we start with an actual first listen or we try to reset our brain to be able to judge the arrangement as a whole. Then we start making adjustments. Here is where the gloves are off and there are no more rules. The differences between the many approaches that different engineers have, are pretty stark. Some start with the vocals, some start with the guitars, some with the drums etc. There is not set way to do this, but the overall goal is still the same. Find the important elements of the piece and make them sound as amazing as you can with all the tools that you might need for it. Then identify the less important elements and reduce them to their essence, so they can still serve their purpose to the song, but leave enough room for the important ones. For these tasks we use dynamic, spectral- and time-based effects to alter the sonic properties of the individual tracks so they mix well together and help convey the feeling and message of the song.
With the abilities of modern plugins and the extreme amount of control the engineer has over every single element, it becomes very obvious why good mixing engineers are so well regarded in the world of music. But at the same they are not magicians. A great arrangement and a great recording can easily be completely destroyed by a mix. But a bad song and bad recording will never be more then that even if the most talented mixing engineer works on it.