Hi vfoley74. Welcome to the forums.
The most trendy thing to do these days is what Stevie D said, using a "lightbox" effect where a larger version of the video pops up over the page. (It's not quite the same as the 19990s thing you mentioned of opening a new popup page.) Personally, though, I'm not totally sold on that lightbox method. You'd want to make sure the video still plays with JS off ... and I also get nervous about these effects in terms of accessibility (those not using a mouse etc.)
Personally, my preference is to have the video on the page and playable on the page, with the option to play the video at full screen size—which most video players seem to have these days.