Mindfulness is the state of acknowledging what is happening in the present moment and accepting it as it is, without judging or evaluating it.
Our every moment is usually filled with thoughts - assessing, judging, worrying, regretting, protesting. We ruminate about the past and worry about the future. There is little space left for attending to the present moment as it is. That way, we identify with our thoughts and don't see ourselves as separate from them. Our thoughts become our reality, and we become slaves to the thoughts that are rapidly generated by our brain and interpreted by our mind.
If, however, you stop and attend to the present as an observer, without identifying with the thoughts and the judgement, you develop a capacity to observe what your mind is doing. That way, you gradually learn to mindfully respond to situations and thoughts, as opposed to being mindlessly reactive to them.
There are two main approaches to developing mindfulness - formal meditation practice and informal mindfulness of the moment practice.
Meditation teaches you to attend to present moment by bringing your awareness back gently from external and internal distraction. It requires allocating some time in your day, starting with a few minutes, to quietly sit and breathe, while acknowledging your thoughts, feelings and physical reactions, but not engaging with them.
The informal practice involves disengaging from your reactive thoughts as you go about your day. This does not mean that your mind needs to be "clear of the thoughts" and does not involve thought stopping. It just means acknowledging the thoughts, accepting them as they are, and continuing to attend to the present moment.
By becoming mindful, we develop an an ability for conscious living, being in touch with yourself, and seeing things as they are.