If you want to show a thumbnail of a picture from the Geograph collection, just copy and paste its URL (web address) between a pair of square brackets: [http://www.geograph.org.uk/photo/1234567] or just its id number between a pair of triple square brackets: [[[1234567]]].
You can't generally show pictures from other websites, but you can link them so people can click the link to view the picture elsewhere by using this syntax: [url=http://www.example.com/pic.jpg]anchor text[/url] where 'anchor text' stands for the text people click to go to the picture.
If you've got a picture that adds value to the point you're making but it's not suitable to upload it to the Geograph collection, you can upload it to the Geograph media server http://media.geograph.org.uk/ . Once uploaded you can show it in a forum thread using the [img]http://media.geograph.org.uk/files/....[/img] syntax. Use the 'direct link' URL from the upload page for this. This works for png, gif and jpeg formats. If the image is wider than 640px, it'll automatically be shrunk to fit.
This technique is best used for supporting material such as out-of-copyright photos that you can't license under the Creative Commons licence, user-generated maps or graphs showing statistics etc. It should not be used to create signatures, smileys or avatars or other recurrent images, which aren't used on the Geograph forums.