Some Twitter posts show a just link and other times the link has an image card. Why is this?

The linked page must be configured properly to contain links to image files for use on social media. For further information see http://ogp.me/.

If you have a wordpress site and the Yoast plugin installed you can use this to set default images for pages, post and different social platforms.

The FB image is on by default the Twitter image needs enabling in the Yoast social settings