A quote is a forum function to draw attention to another player's post in whole or in part, either to pick out part of a post or to specify which post is being replied to. This is generally displayed as a box containing the quoted post, with a link to the original.
Quoting is a fundemental element of forum mafia. It allows players to respond to segments of posts without nebulous references to "the fifth line of post 184". However, excessive use of quotes can be considered annoying.
A quotewall is when a player quotes a very large portion of another's wallpost. Since quotes are supposed to point out specific things, quoting a whole wall is frowned upon because it often adds no value to the post, in return for adding much text to the page.
An in-quote response is when a player breaks up a quotewall by interspersing their own responses and commentary, often in bold or coloured text. This helps break up the wall but leaves the quotebox around everything, which can make the new post harder to respond to.
Quotestripes are an alternative to in-quote responses, where one quote is broken up into a series of smaller ones, their boxes causing a stripe effect across the page.
Should two or more players quote each others' quotes as the discussion continues, a quote pyramid is formed, named for how the innermost quotes are ever more indented. Quote pyramids are much-hated by many players due to the screenspace they bloat up for diminishing benefit (the taller a pyramid is, the less important the inner/early quotes are).