In a WordPress post, you can simply place the URL from a YouTube video or tweet into your editor, and the video or tweet will automatically be embedded.
You can also do this with media from a number of other sites such as Flickr and Vimeo and Slideshare. (See the entire list here.)
Trying to do that in a text widget, however, doesn’t work.
But there’s a way around that.
Code Snippet for Embeds in Text Widgets
As you’ll be changing your theme for solution, you should probably consider either creating a child theme or making your own simple plugin to use.