As a pre requisite, you must have a Flickr account. I recommend having a Pro account (only $25 / yr)
Once you have an account and are signed in, you need to create an API key. You can get this by going to the /account section of Flickr
Click on the "Sharing & Extending" tab
Under your API keys, create a new API key and write down the Key and Secret somewhere safe.
Next, download, install, and enable the FlickrGallery module
After you enable the module, in Drupal go to the configuration page at /admin/config/media/flickr
Chances are, you will need to download and place phpFlickr.php inside sites/all/libraries/phpFlickr:
http://code.google.com/p/phpflickr/ (Drupal will give you a warning if so)
You'll also need your Flickr ID, which you can conveniently get from here:
This is the screen you should see that will allow you configure your Flickr integration:
Make sure you download and enable the lightbox2 module
Then change the settings under Flickr Settings and modify the CSS styling to fine-tune the output display.
You can watch my demo, which starts off by explaining how to setup Flickr in Drupal 6 using the media and media-flickr modules, which allows for a more fine-tuned Flickr integration, but is also significantly more difficult to setup.