Mindfulness is simply a particular way of paying attention, however, there are some formal meditation practices that can help us to strengthen our capacity for it. Mindfulness is a very simple form of meditation that was little known in the West until recently. A typical meditation consists of focusing your full attention on your breath as it flows in and out of your body. Focusing on each breath in this way allows you to observe your thoughts as they arise in your mind, and, little by little, to let go of your struggle with them. You come to realise that thoughts come and go of their own accord, that you are not your thoughts. You can watch as they appear in your mind, seemingly out of thin air, and watch again as they disappear, like a soap bubble bursting. You come to the profound understanding that thoughts and feelings (including negative ones) are transient. They come and they go, and ultimately, you have a choice about whether to act on them or not.
I am a trained mindfulness teacher and have run courses in Mindfulness Based Cognitive Therapy in a hospital setting. Whilst teaching formal mindfulness meditation practice will not necessarily be a part of every psychotherapy, for many people I would encourage practising between sessions as a useful adjunct to the therapy.