I personally recommend the following:

Screen Capture

- OBS - Free, but consider donating if you have the means. Great software! Very popular right now.
- FRAPS - Costs $37. Been around a long time. Records desktop and games. Pretty popular also.

Video Production Software

- Sony Movie Studio - I use it and I think it's a great value for $60. Tons of features for only $60. Does almost anything an amateur video producer would want.

Video Production and Screen Capture
- Camtasia - $300 but more robust all-in-one software solution that includes comprehensive video production tools and screen capture. Generally targeted at enterprise/business audience that wants to make training videos, but good for recording webgames, too. I've found this is only good for desktop recording. I can't use this to record PC games, like Skyrim or whatever.

Here's a reddit guide to making webms using OBS and XMedia Recode which are both free software. XMedia Recode allows you to convert to any video format, trim videos/audio, etc and it's free!
It takes me about 2-3 minutes to record, trim, convert, and upload a short(<15 sec) webm.