I will try to address both ways this question could be interpreted - counting total number of views and counting one view.
Counting Views:
When a video is uploaded, it is cached in the different YouTube servers spread around the globe. This is done, of course, for faster response times. Every time a request is made for a video, the closest server provides the video and increments the count in its local log. These logs are then collated at fixed times and a central log is then updated.
The 301 freeze: When a video has less than 300 views, the views are not verified. However, when views cross 300, YouTube starts verifying them to make sure that they are legitimate playback requests and not "fake" views. So you would often find that the count of a video gets stuck for a few hours or a day at 301 (sometimes 302 or 305 or 308 if multiple logs sync with the centralized server at the same instant). From this point onwards, view counts increase in batches - after every batch of new views are verified by YouTube.
An interesting technical tidbit is that when the number 300 was decided by the designers, the programmer made the elementary careless mistake of using the age old ViewCount <= 300 instead of the ViewCount < 300 in their logic.
Calculating a view:
Coming to the specifics of the question, what constitutes a view? View is a video playback that was requested by an actual user. YouTube has been extremely tight lipped about this because giving away details of even the minutest of mechanisms will cause people to "hack" the system and artificially inflate views.
However, we can make some guesses by observation. We know that fake views include misleading views - misleading titles and thumbnails that attract views. When a video has a large number of views that last for mere seconds after clicking, the views are not counted as legitimate. So if a video is viewed in its entirety by someone who clicked on it, it is counted as one view. But not all views are fully played.
Google Ad Sense works only with videos that are over 30 seconds in length so that the click through rates get registered. In fact, some videos are lucky enough to have just ten seconds of play considered as a view. We can conclude from this that the amount of video played should be above a threshold percentage of the length of the video. The type and genre of video could also be a factor. YouTube also considers views from the same IP in breaks of 6 to 8 hours. So one person viewing the same video repeatedly would only generate 3 to 5 views a day, after views cross 300. A viewer being redirected to YouTube upon clicking an embedded video counts as one view. If there is an embedded video with autoplay, it is not counted as a view. I'm quite sure there are many more restrictions and rules that go into categorizing a request as a view that we might not be aware of.
In conclusion, a view is a unique, user-initiated play of a certain secret percentage of a video that is played only by YouTube's hosting.
