Whether you're using Joomla's standard articles to manage content, or Joomlaworks' K2 advanced content extension, a fantastic (and free) plug-in for adding videos is Joomlaworks' AllVideos. This extension is simple to install, includes a "Responsive" option that automatically resizes the video to your website's layout, and is easy to use.
Once you've installed and enabled the extension, simply add the plug-in's descriptive tags and the video's unique ID and you're done! In the following example, we're using ASAPScience's "Fresh vs. Frozen" video to demonstrate. The following is the sample code that we've added in the page's WYSIWYG editor (keep in mind that the highlighted code is normally hidden and replaced by the actual video), followed by the end result. In this case, the video provider is YouTube, and the video's unique ID is "zjsOOT347cA".
If you're using K2, simply click on the "Media" tab when editing the K2 Item, select the media source you'd like to use (options include upload, browse your server, select an online source, or embedding code) and you're done!
Here's an example of the code when using Vimeo as the provider:
AllVideos support a number of video providers, and also supports the popular provider "SoundCloud" for those that would like to include audio in their content.