What is the single word that describes a person who always gossips? (For example, telling things about others which are not true.)
Er... perhaps a gossip? You may also see gossipmonger and gossiper, but in US English just plain old gossip is the standard noun.
As onomatomaniak said, a person who habitually spreads rumors or engages in indiscreet talk (ie. gossips) can be referred to as a gossip. One thing to note, this would be independent of whether the things he or she tells are true or not.
Alternatively, you could also describe such a person as a rumourmonger, or even a scandalmonger, depending on the nature of the gossip and the intent.
One might call such a person a quidnunc, although that might be thought somewhat out-of-date nowadays. Merriam-Webster suggests quidnunc is more likely to be someone keen on hearing the latest gossip than repeating it, but I have never met anyone who just listened to scandal and rumour without passing it on.
Taleteller: One who tells tales or stories, especially in a mischievous or officious manner; a talebearer; a telltale; a tattler.
Slanderer: One who makes false and injurious statements about another, one who slanders, one who defames, one who maligns.
Backbiter: One who slanders, calumniates, or speaks ill of the absent; person who says nasty things about another person behind the second person's back: that is, out of sight and hearing of the second person.
One who tattles (reports others' wrongdoings), often by a child seeking attention. (Wiktionary)
P.S. You did say that the person always gossips, whether what they say is true or not. The fact that they do it a lot could mean that the person thrives on it, and suggests that it might be child-like, attention-seeking behavior.