Animated shows produced for the Direct-to-Video market, almost universally abbreviated "OVA" or "OAV" (for "original(ly) animated (for) video"), and more rarely OAD (same for DVD). The term is almost always associated with anime; however, this form of storytelling is beginning to become popular in the anglosphere with recent animation based on comic characters and TV shows. Unfortunately, these still labour under the shadow of the term "direct-to-video".
While the terms "direct-to-video" and "direct-to-DVD" have negative overtones, essentially being synonyms for Shovelware in the United States, "OVA" has almost exactly the opposite connotation. This partially arises from the view that Western "direct-to-video" releases are not good enough (or too explicit) for theaters; OVAs, on the other hand, are seen as a step up from regular television production. Because the production house need not adhere to the rapid-fire schedule or constrained budget of a TV series or feature-length film, more effort and care can be applied to an OVA, resulting in a much higher level of quality. Additionally, since OVAs aren't aired to the public, Getting Crap Past the Radar is a moot point, which allows shows for older audiences to avoid ducking more mature subject matter. On the flip side, the vast majority of hentai series are produced and released as OVAs, just like much live-action pornography is released directly to home media or online. Given the general lack of a clear-cut production schedule, the time duration of an OVA is rather varied— some are 26 minutes long, while others are 60-80 minutes, but the idea of an OVA being up to two hours long is uncommon.
There are some caveats to the increased freedom, though. OVAs are often produced "on speculation", with no guarantee that the story they tell will ever be completed— and many are not. At least one OVA series ends with a plaintive plea for more money so the creators can continue making the show. However, even this is not always a guarantee— the Hellsing OVA series managed to adapt the entire ten volume manga it was based on.
One trend which has become evident recently is the continuation of broadcast television series in OVA form after they complete their initial run; the aforementioned lack of broadcast standards also allows writers to work in anything they couldn't put into the original TV show. Inversely, the exposure of a broadcast initial run may be a lure for viewer interest in the less censored, more serious story continuing on home media. Due to the heavy market decline of physical media in the west during the late 2000's and early 2010's, both the broadcast and OVA-exclusive material are typically carried over via streaming services, thus removing the physical distinction between them (said decline was also responsible for the heavy downturn in the animated hentai industry during the 2010's, which previously used the western DVD boom to accrue more money and support more technically ambitious projects).
OVAs were most common in the 1980s and 1990s, particularly during the 1986-1991 asset price bubble (which, up until its bursting the following year, resulted in studios having a lot more money to back up the prospect of huge creative freedom), during which many well-known series were released in this fashion. The idea of this era as "the Golden Age of Anime" stems in part from the OVA boom, which led to a large number of unique stories being produced unrestricted and without a specific target audience and brought a level of creative freedom comparable to New Hollywood's effects on the American film industry. With the recent rise of 12-Episode Anime series as an alternate short format, as well as streaming video over the Internet, OVAs have come to be less frequently released, though by no means extinct; the aforementioned Hellsing, Code Geass: Akito the Exiled and Ghost in the Shell: Arise are a sampling of recent OVA releases. Anime made for release on the Internet are called Original Net Animation, or ONA for short, and act as a Spiritual Successor of sorts to the OVA format (with some works, like the 2002 Azumanga Daioh anime adaptation, using the ONA format to gauge audience interest with a pilot).