What is a signature? A signature is text, images, etc. that appear below all of your posts.
To edit your signature go to the User CP, and click on Edit Signature, located in the left-hand column. There, you can edit your signature. You may put whatever you wish there. Some common features of signatures include your name/username, a picture of your budgie(s) with their name(s) on it, or a quote. However, you may put whatever you wish, as long as what you put still follows the rules (no links to competing sites, no spam, etc.).
If you don't want to include your signature in a single post of yours, simply uncheck the Show your signature box located under the editing box if you're using quick reply, or in the Additional Options box if you're creating a thread or using full reply.
If you don't want to view others' signatures go to your User CP, click on Edit Option (in the left-hand column), scroll down to the third box titled Thread Display Options, and uncheck the Show Signatures box. This will disable the signatures for everyone so you will not see anyone's signatures when viewing threads.