Many people use YouTube videos on their websites for a better representation of their content. Some websites provide a link to YouTube video and some websites provide an embedded link of the same.
If a link is provided then user is redirected to Youtube but when you use a embed code to display the video on your site, the video itself appears on the webpage. The user remains on your site, where it only appeared as a link.
A lot of people says that it is not legal to use someone's work on your webssite without their permission and it is a copyright violation.
As per youtube terms and conditions:
License to YouTube
By providing Content to the Service, you grant to YouTube a worldwide, non-exclusive, royalty-free, transferable, sublicensable license to use that Content (including to reproduce, distribute, prepare derivative works, display and perform it). YouTube may only use that Content in connection with the Service and YouTube’s (and its successors’ and Affiliates) business, including for the purpose of promoting and redistributing part or all of the Service.
License to Other Users
You also grant each other user of the Service a worldwide, non-exclusive, royalty-free license to access your Content through the Service, and to use that Content, including to reproduce, distribute, prepare derivative works, display and perform it, only as enabled by a feature of the Service (such as video playback or embeds). For clarity, this license does not grant any rights or permissions for a user to make use of your Content independent of the Service.
When any user uploads a video, he can either enable or disable embedding, If he has left it enabled, that means he grants the user a limited license to embed the video.
You may use the Service as made available to you, as long as you comply with this Agreement and applicable law. You may view or listen to Content for your personal, non-commercial use. You may also show YouTube videos through the embeddable YouTube player
Fo details: https://www.youtube.com/static?template=terms
So it seems that embedding a YouTube video is legal, as long as the video being embedded is not an infringing video in itself.