Encode settings for HTML5 background video
You should export multiple versions to accommodate for different browsers. With HTML5, you can offer different versions of the video and the browser will automatically pick the first one that's supported. For example, take a look at the source code of the paypal website you refered to:

&lt;video autoplay="autoplay" muted="muted" poster="none"&gt;
    &lt;source src="https://www.paypalobjects.com/webstatic/mktg/wright/videos/send-money.mp4" type="video/mp4"&gt;
    &lt;source src="https://www.paypalobjects.com/webstatic/mktg/wright/videos/send-money.webm" type="video/webm"&gt;
&lt;/video&gt;


As you can see, they offer both a mp4 (H264 codec) and a webm version of the video. Those would be good options for you as well. 

Furthermore, you should try to keep the videos as small as possible, so that site visitors with slow internet will also be able to watch it. For that purpose, you should export the video with 720p instead of 1080p (nobody will see the difference, especially if the video is in the background) and aim for a bitrate below 1 Mbit/s. For example, the paypal video uses a bitrate of about 700 Kbit/s. If you don't need the audio, you could export without audio or with a low audio bitrate, i.e. 96 Kbit/s.

You can also save the video from the website to your computer and check the codec, bitrate and other parameters e.g. with MediaInfo or similar programs.

Update November 2018

Since this question is still getting a lot of views, I wanted to add some up-to-date information. By now, all modern browsers support MP4 (MPEG-4/H.264), and have been for a long enough time that you will be fine offering only MP4 for most projects. Check the link above for a browser compatibity table to make sure all browsers your project needs to support can play MP4. Keep in mind to use AAC for the audio track, not MP3, since it's supported by fewer browsers.

Still, you can and should offer a WebM video as the first &lt;source&gt;, since it features better compression, so users on a limited data plan will thank you. I just wanted to add that if you are limited in terms of disk space, render time or some other variable that makes it difficult to offer WebM, MP4 is all you need by now. 

Also, since my original answer included something about audio, one additional takeaway that's not strictly related: Browser vendors are cracking down on autoplay videos (video that start playing automatically after loading the page, without a user interaction), especially if the video contains audio. For example, Safari will only allow autoplay videos if the video contains no audio track, or the video is muted (using the corresponding attribute on the &lt;video&gt; element). So if you don't need the audio track, export your video without one (autoplaying videos with audio are obnoxious UX anyway). And if you do need the audio, keep in mind that some browsers will block the autoplay, so make sure to include a placeholder image using the poster attribute.