How do I get a YouTube video thumbnail from the YouTube API?

What Asaph said is right. However, not every YouTube video contains all nine thumbnails. Also, the thumbnails' image sizes depends on the video (the numbers below are based on one). There are some thumbnails guaranteed to exist:

Width | Height | URL
------|--------|----
120   | 90     | https://i.ytimg.com/vi/<VIDEO ID>/1.jpg
120   | 90     | https://i.ytimg.com/vi/<VIDEO ID>/2.jpg
120   | 90     | https://i.ytimg.com/vi/<VIDEO ID>/3.jpg
120   | 90     | https://i.ytimg.com/vi/<VIDEO ID>/default.jpg
320   | 180    | https://i.ytimg.com/vi/<VIDEO ID>/mq1.jpg
320   | 180    | https://i.ytimg.com/vi/<VIDEO ID>/mq2.jpg
320   | 180    | https://i.ytimg.com/vi/<VIDEO ID>/mq3.jpg
320   | 180    | https://i.ytimg.com/vi/<VIDEO ID>/mqdefault.jpg
480   | 360    | https://i.ytimg.com/vi/<VIDEO ID>/0.jpg
480   | 360    | https://i.ytimg.com/vi/<VIDEO ID>/hq1.jpg
480   | 360    | https://i.ytimg.com/vi/<VIDEO ID>/hq2.jpg
480   | 360    | https://i.ytimg.com/vi/<VIDEO ID>/hq3.jpg
480   | 360    | https://i.ytimg.com/vi/<VIDEO ID>/hqdefault.jpg

Additionally, the some other thumbnails may or may not exist. Their presence is probably based on whether the video is high-quality.

Width | Height | URL
------|--------|----
640   | 480    | https://i.ytimg.com/vi/<VIDEO ID>/sd1.jpg
640   | 480    | https://i.ytimg.com/vi/<VIDEO ID>/sd2.jpg
640   | 480    | https://i.ytimg.com/vi/<VIDEO ID>/sd3.jpg
640   | 480    | https://i.ytimg.com/vi/<VIDEO ID>/sddefault.jpg
1280  | 720    | https://i.ytimg.com/vi/<VIDEO ID>/hq720.jpg
1920  | 1080   | https://i.ytimg.com/vi/<VIDEO ID>/maxresdefault.jpg

You can find JavaScript and PHP scripts to retrieve thumbnails and other YouTube information in:

• How to get YouTube Video Info with PHP
• Retrieve YouTube Video Details using JavaScript - JSON & API v2

You can also use the YouTube Video Information Generator tool to get all the information about a YouTube video by submitting a URL or video id.

In YouTube API V3 we can also use these URLs for obtaining thumbnails... They are classified based on their quality.

https://i1.ytimg.com/vi/<insert-youtube-video-id-here>/default.jpg -   default
https://i1.ytimg.com/vi/<insert-youtube-video-id-here>/mqdefault.jpg - medium 
https://i1.ytimg.com/vi/<insert-youtube-video-id-here>/hqdefault.jpg - high
https://i1.ytimg.com/vi/<insert-youtube-video-id-here>/sddefault.jpg - standard

And for the maximum resolution..

https://i1.ytimg.com/vi/<insert-youtube-video-id-here>/maxresdefault.jpg

One advantage of these URLs over the URLs in the first answer is that these URLs don't get blocked by firewalls.

Each YouTube video has four generated images. They are predictably formatted as follows:

https://img.youtube.com/vi/<insert-youtube-video-id-here>/0.jpg
https://img.youtube.com/vi/<insert-youtube-video-id-here>/1.jpg
https://img.youtube.com/vi/<insert-youtube-video-id-here>/2.jpg
https://img.youtube.com/vi/<insert-youtube-video-id-here>/3.jpg

The first one in the list is a full size image and others are thumbnail images. The default thumbnail image (i.e., one of 1.jpg, 2.jpg, 3.jpg) is:

https://img.youtube.com/vi/<insert-youtube-video-id-here>/default.jpg

For the high quality version of the thumbnail use a URL similar to this:

https://img.youtube.com/vi/<insert-youtube-video-id-here>/hqdefault.jpg

There is also a medium quality version of the thumbnail, using a URL similar to the HQ:

https://img.youtube.com/vi/<insert-youtube-video-id-here>/mqdefault.jpg

For the standard definition version of the thumbnail, use a URL similar to this:

https://img.youtube.com/vi/<insert-youtube-video-id-here>/sddefault.jpg

For the maximum resolution version of the thumbnail use a URL similar to this:

https://img.youtube.com/vi/<insert-youtube-video-id-here>/maxresdefault.jpg

All of the above URLs are available over HTTP too. Additionally, the slightly shorter hostname i3.ytimg.com works in place of img.youtube.com in the example URLs above.

Alternatively, you can use the YouTube Data API (v3) to get thumbnail images.

You can use YouTube Data API to retrieve video thumbnails, caption, description, rating, statistics and more. API version 3 requires a key*. Obtain the key and create a videos: list request:

https://www.googleapis.com/youtube/v3/videos?key=YOUR_API_KEY&part=snippet&id=VIDEO_ID

Example PHP Code

$data = file_get_contents("https://www.googleapis.com/youtube/v3/videos?key=YOUR_API_KEY&part=snippet&id=T0Jqdjbed40");
$json = json_decode($data);
var_dump($json->items[0]->snippet->thumbnails);

Output

object(stdClass)#5 (5) {
  ["default"]=>
  object(stdClass)#6 (3) {
    ["url"]=>
    string(46) "https://i.ytimg.com/vi/T0Jqdjbed40/default.jpg"
    ["width"]=>
    int(120)
    ["height"]=>
    int(90)
  }
  ["medium"]=>
  object(stdClass)#7 (3) {
    ["url"]=>
    string(48) "https://i.ytimg.com/vi/T0Jqdjbed40/mqdefault.jpg"
    ["width"]=>
    int(320)
    ["height"]=>
    int(180)
  }
  ["high"]=>
  object(stdClass)#8 (3) {
    ["url"]=>
    string(48) "https://i.ytimg.com/vi/T0Jqdjbed40/hqdefault.jpg"
    ["width"]=>
    int(480)
    ["height"]=>
    int(360)
  }
  ["standard"]=>
  object(stdClass)#9 (3) {
    ["url"]=>
    string(48) "https://i.ytimg.com/vi/T0Jqdjbed40/sddefault.jpg"
    ["width"]=>
    int(640)
    ["height"]=>
    int(480)
  }
  ["maxres"]=>
  object(stdClass)#10 (3) {
    ["url"]=>
    string(52) "https://i.ytimg.com/vi/T0Jqdjbed40/maxresdefault.jpg"
    ["width"]=>
    int(1280)
    ["height"]=>
    int(720)
  }
}

* Not only that you need a key, you might be asked for billing information depending on the number of API requests you plan to make. However, few thousand requests per day are free.

Source article.