Add Reply - this is the basic ability to add a reply. If you click this image button,
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you will be brought to a full posting form where you can add a reply to the topic with all of the basic posting options.
Fast Reply - this is a much quicker ability to reply to a topic for users who do not need to use all of the advanced posting options. If you click this image button,
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a fast reply box will expand below the topic (if you configure your account to automatically expand fast reply boxes in your user control panel under the Board Settings page, this will already be expanded) and you can type in a reply to the topic and submit it. You will not have easy access to bbcode options, but you will have the ability to open a smilies popup window to insert smilies.
Tip: You can also highlight text in the web page, scroll down to your fast reply box and hover over it - a new button will appear saying "Add selected text as quote" allowing you to easily quote text on the same page.
Multi Quote - the multi quote button,
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is actually used for multi-quoting capabilities. You can click the multi quote button on several posts, and then click the "Add Reply" button at the bottom of the page. When you do this, you will be presented with a full posting form (as outlined in example 1 above), however you will also have the contents of each of the posts you selected to quote inserted into your posting form with "quote" bbcode tags wrapping them. This way if you wish to quote 3 or 4 other posts in one reply, you can select the posts to quote, go to the form, and add your reply under their quoted text.
Quote - this works similar to the multi quote button,
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except it is designed to allow you to quote one post quicker. When you hit quote underneath a post, you will be brought to the full posting form (as in example 1) with that one post wrapped in quote tags. You can acheive the same result by clicking the multi quote button on that post, and clicking Add Reply, but why click two buttons if you only need to click one right?
Hopefully this clarifies how these buttons are intended to be used, and their exact behavior, for anyone confused by their functionality.