Pure O is a common informal term used by OCD sufferers to refer to what a psychologist would have once called Pure Obsessional OCD and would now usually call Primarily Obsessional OCD.
Pure O is OCD without obvious, overt compulsive behaviors. Someone with Pure O will have frequent intrusive thoughts (obsessions), and respond to them compulsively, but will not have the major behaviors we generally associate with OCD. Instead, the responses will be mental rituals and/or non-apparent compulsive behaviors, such as avoiding certain places, people, or actions, fixating on certain topics of conversation, etc.
The most commonly referred to symptoms are intrusive, unwanted thoughts about stressful subjects. These may be persistent thought of self-harm, fears of hurting others, fear of sinning, persistent 'sinful thoughts', and irrational fears that one might be a sexual deviant -- e.g. fears that one is exhibiting homosexual tendencies or is at risk of molesting a child. These later two are so common in modern Western culture that they are often referred to specifically as Homosexual OCD (HOCD) and Pedophile OCD (POCD).
On the other hand, some obsessive thoughts would not be stressful except for the fact that they are persistent. These include intrusive and obsessive awareness of automatic bodily functions, such as breathing, swallowing, or blinking.
It is important to remember that everyone has annoying thoughts of various sorts, including those mentioned above. Pure O needs to be diagnosed by a professional, and will generally only be diagnosed when obsessive thoughts are intrusive and persistent, have a strong negative effect, and are given undue weight in a person's mind. Other disorders that may by confused with OCD and Pure O include obsessive-compulsive personality disorder (OCPD), autism spectrum disorders, generalized anxiety disorder, or a true phobia.
Sources and further reading:
OCD Center of LA: Pure Obsessional OCD
Reddit: r/OCD: Pure O discussion
Wikipedia.org: Primarily Obsessional OCD