There's a lot of fan-generated Star Trek videos out there, and most of it isn't "production quality". There are, however, a couple of organizations that are doing top-notch work.
Star Trek Continues looks like they film on the original series Star Trek set. The writing, the stories, the characters--it all looks straight from the hand of Roddenberry. The acting is so reminiscent of the original series characters that the only difference is the fact that the actors aren't Shatner, Nimoy, et. al.
Watch these full-length episodes:
Pilgrim of Eternity
Fairest of them All
The White Iris
Even better, 2016 promises a full-length independent motion picture, Axanar. First watch the 21 minute teaser video, Prelude to Axanar. Then watch the feature film teaser for Axanar; the full movie is expected to be released in late 2016. Here is the indiegogo fundraising page if you'd like to contribute.
I have to agree with one of the commenters on the feature film teaser video: "Ok. So 'Fan Film' now means 'A film so wonderfully produced that fans
will love it more than the actual franchise?' Sure. That sounds cool."
Now seriously, go watch Prelude to Axanar.