The full episodes of "Days of Our Lives" are available on NBC's website as of October 2015, but they are available only for a short time frame, typically a week after air date. All of the previous episodes only give brief descriptions and a highlight video about that particular episode.
To watch these episodes, go to the NBC home page, and click on the Full Episodes link. Next, click on "Days of Our Lives" under Current Episodes. The next page shows five recent full episodes along with other Highlight videos. Click on the full episode you are interested in watching.