To efficiently and easily produce and publish a short video for teaching-support purposes (for example, to answer a student's question from remote, or for clarifying a difficult concept), I typically go through the following light-weight process:
a) a working webcam
b) an account on www.hipcast.com
1. prepare your flow of thought, and - if necessary - readable paper sketches as support
2. log in to hipcast, turn on your webcam and start recording "online" your video
3. review and save your video
4. publish it through hipcast by sending it to your blog (e.g. on blogger)
5. if you want to have it visible on your blog, fine. Otherwise, if you want to use it as "mesh-up" anywhere else, copy from the blog post the generated code and paste it in the place to want to embed it (in a forum, or in a webpage or in a HTML mail message that you can send to your students).
6. the job is done! wait for your students' feedback.