Creating posts in Steemit User Interface:
option 1) in WYSIWYG (What You See Is What You Get) mode: press the link button at the top of your editor and paste the URL in; YouTube is automatically embedded
option 2 use the Markdown editor in Steemit UI. the format of a link under a piece of text is [text] (URL) without a space between the ] and ( (needed to include the space, otherwise you would not see the (URL) part.

Note1: Comments are always markdown style.
Note2: when you like to include some offline or even online images, you can copy/paste them in the Steemit Editor. Steemit uploads this to its own media storage automatically.

When you like to get media into your post and have them playable like a YouTube movie, or Soundcloud recording, you may like to copy/paste the 'Embedded Code' these site offer you when clicking the 'Share' button of the media you like to share; This embeds the media a nicer way than the URL does. Some sites like Mixcloud offers Embedded code as well, but Steemit does not support them; The reason for this I dont know, but it is easy to check.