Obsessions are thoughts that are repetitive, intrusive, involuntary, irrational, and anxiety-provoking. Any thought that has these five criteria can be considered obsessions.
When people commonly talk about being “obsessed” with something — for instance, “Scott is obsessed with dating Julie” — they mean that Scott thinks repeatedly about Julie. Perhaps thoughts of Julie intrude into his consciousness from time to time. But I would doubt that Scott would describe his thoughts about Julie as irrational, or involuntary, or anxiety-provoking (unless he has a touch of social anxiety or fears rejection). Scott’s condition would better be described as an “infatuation.” (One should note that the word originally meant “to be made foolish.”)
Obsessions also need to be distinguished from worries, although the distinction here is more murky. The essential difference is that obsessions are irrational and thus are not concerning real-life problems. The person who obsesses may know that the stove is turned off, or that they didn’t accidentally drive over someone, but still they have to check in order to feel certain. The person with obsessions would also want to think of anything except their obsession.
Worrying, on the other hand, is best defined as dwelling on one’s feelings or problems with reference to the future; if the reference is to the past, this same process is called rumination. Worry and rumination are not entirely involuntary: if a person were asked whether worrying or ruminating does them any good, they would often say yes (while admitting that it also does them harm). The person with OCD would generally describe their obsessions as completely useless and would gladly be free of them in a heartbeat. Worriers find it harder to want to give up worrying.
Finally, it is important to know that just having obsessions does not equal having obsessive-compulsive disorder (OCD). Eighty percent of the population has occasional obsessive thoughts. What defines OCD is the disordered fear of obsessions, which causes great distress, and may be accompanied by compulsive actions that artificially terminate the obsessive thought. Generally speaking, the less willing one is to have obsessions, the more likely one is to have them and to go on to develop OCD. If obsessions are accepted as quirky products of one’s association cortex, and are given no more importance, they will not become feared — and so one would be protected against ever developing OCD.