What is a subconscious mind?
Imagine if I say the word lemon. Do you notice an image, a smell or taste immediately pop up? Perhaps you can feel its texture? That is your subconscious working. Notice how effortless it is.
Your conscious mind is what some call right brained activity, thoughts, language and reason. The subconscious is about feelings, the feelings that are behind reason and words.
An example of a conflict between the conscious and subconscious mind would be somebody that want to do their best in school, but at the same time hate and have to force themselves to study.
In the end the subconscious wins the conflicts over the conscious.
But there are ways for the conscious to take control over the subconscious and get rid these internal conflicts. EMDR is an example.
Whether we like it or not, feelings rule over reason. Feelings are not logical and make no sense to the conscious mind. That is why we may suffer insomnia with the mind going in circle trying to make sense of difficult feelings.