So I keep seeing things like "The Great Wyrm", "Forest of the Wyrm", "Wyrm aged" and they seem to be all related to Dragons. Is Wyrm just another name for Dragons? Is this a general thing or does it change from setting to setting?
Originally, the term is an Old English word that means "serpent" or "snake". It was commonly found in old European poems, where it referred to a wingless dragon. The term was later used to refer to any dragon, as with Tolkien's usage in The Hobbit and other works (which heavily influenced D&D).
In D&D, "wyrm" refers to a large, presumably old, dragon. Third Edition classified True Dragons (but not Lesser Dragons) into age categories, with each category signifying a greater level of size & ability. The smallest category, "wyrmling", represented a young member of the True Dragon race, while the largest, "Great Wyrm", signified a terrible beast of great size and power. Further information about the True Dragon classifications and the size/age/power for each can be found on the D&D wiki at the link below:
Note: As @SevenSidedDie points out in the comments below, these classifications have existed in some form as early as the 1979 Monster Manual. I started with 2nd edition but I've slept a lot since then.
In 1994, TSR released a dragon-specific campaign set called "The Council of Wyrms". IIRC this was the set that released rules for playing dragon (and half-dragon) characters. It also introduced new rules for encountering or fighting dragons.
Nowadays, the term is often found in other fantasy sources as well, although sometimes the meaning is changed so that it refers to a smaller dragon variation (typically smaller than a "drake"). In some cases, a Wyrm is not a proper dragon at all, but rather a similar animal that is related to the dragon (aka "dragonkin"). In D&D, you can find this used for creatures like the "Landwyrm", which is not a proper dragon but is still classified as the dragon type as far as the rules are concerned.
"Wyrm" (and its variant spelling "worm") is a common but old synonym for "dragon" in English. It's not originally a D&D or RPG term, but it's seen more often in fantasy RPGs (and fantasy literature) than everyday English because archaic words lend games a more fantasy feel.
There are two meanings to the "Wyrm" in D&D.
It's just a really big / old dragon
Game mechanics term
In D&D 3.0 / 3.5 Wyrm and Great Wyrm are age categories of True Dragons. As can be seen in SRD:
Category Age (Years) Wyrmling 0-5 Very young 6-15 Young 16-25 Juvenile 26-50 Young adult 51-100 Adult 101-200 Mature adult 201-400 Old 401-600 Very old 601-800 Ancient 801-1,000 Wyrm 1,001-1,200 Great wyrm 1,201 or more
So, at least in this edition, mechanically speaking wyrms are dragons over millennia old, and that's all.
The word for dragon in Germanic mythology and its descendants is worm (Old English: wyrm, Old High German: wurm, Old Norse: ormr), meaning snake or serpent. In Old English, wyrm means "serpent", and draca means "dragon".