The term is used to signify many different things, but in the world of cognitive behavioral therapy (CBT) for obsessive compulsive disorder (OCD), it simply means paying attention to the present moment without judgment or analysis.
The present moment is whatever is going on at any given time. This includes the presence of thoughts, feelings, and sensations, whether wanted or unwanted. The struggle of the OCD sufferer is one in which certain internal experiences (thoughts, etc.) are viewed as unacceptable, whereas others are allowed to pass by without critique. Mindfulness suggests a different perspective on the presence of these internal experiences, that they be simply noticed, not judged or pushed away.
Obsessing is the state of being stuck in the mind, being distracted by unwanted thoughts and feelings, but not feeling capable of returning from looking them without compulsions or judgments. One tool for developing the skill of coming back is meditation, in which you attend to an anchor (often breathing), notice when your attention has wandered, and then gently invite yourself back to your anchor. This skill can become more automatic and applied to the experience of intrusive thoughts in OCD.
At the core of mindfulness for OCD is identifying thoughts as simply being thoughts, not threats, meaning the content of the thought itself holds no intrinsic value. Similarly, if you suffer from OCD, you may have come to believe that feelings are facts, signs that your obsessions hold some important truths. Mindfulness challenges this by simply identifying feeling as feelings, experiences that can be observed as they pass through. A common metaphor for this concept is that of sitting on a train platform, watching trains go by without attempting to board them.