Flickr makes it very easy to save on bandwidth when it comes to displaying your photos on other websites. Not only do Flickr provide a robust hosting platform allowing your images to be served quickly and reliably to users all over the world, but there’s no charge for doing so even if it’s viewed a massive number of times.
In return for allowing people to do this, Flickr state, very clearly, that any images hosted by Flickr (regardless of whether it’s your image or not - that’s not relevant) and posted elsewhere should be linked back to the Flickr photo page just like this one,

On the Flickr photo page, when you click to view all sizes, you’re presented with two options; option 1 is some HTML code that will display the image on an external website and link it back to the photo page. Option 2, is the actual image URL. Underneath both of these options is this,
Flickr Community Guidelines specify that if you post a Flickr photo on an external website, the photo must link back to its photo page. (So, use Option 1.)
The following comes from the Flickr Community Guidelines,
Do link back to Flickr when you post your Flickr content elsewhere.
The Flickr service makes it possible to post content hosted on Flickr to outside web sites. However, pages on other web sites that display content hosted on flickr.com must provide a link from each photo or video back to its page on Flickr.
Pretty straightforward, yes?
Here’s one explanation of why you should link your embedded images back to the Flickr photo page from an employee,
The link-back-to-Flickr rule is there as part of a trade off: we allow people to reference image files hosted on our servers and pay for the bandwidth in exchange for some exposure.
So, there you have it. If you embed Flickr hosted images onto external websites (i.e. not Flickr) without linking the photo back to the photo page then you’re breaking the guidelines, taking without paying or in other words stealing.
In fact, you’d be no better than someone who hotlinks to other peoples images without permission and even has the cheek to serve the images from the owner’s website so it uses up the bandwidth that they’re paying for.
If you want to post your Flickr hosted images into a forum, the HTML code provided in option 1 probably isn’t going to work as HTML is usually stripped out of forum posts for a number of reasons. Usually, the alternative is to use forum BB code which may allow you to post images depending on the rules of that particular forum.
To make it a little bit easier to use the HTML code from option 1, I’ve coded up a little tool to convert Flickr embedded Image option 1 HTML to BB Code. The converted code ought to work on the most popular forum systems. Let me know if it doesn’t and I’ll see about changing it.