Well, starting with the easiest question first. OVA stands for "Original Video Animation." But what does "Original Video Animation" means?! It makes little sense in English, obviously, because animations are often original, and they are also in video, so wtf is tautological crap? Well, it's because this is actually a Japanese word in disguise!
It usually happens that Japanese loans words from English, and "OVA," or oobuiee オーブイエー , is one of these cases. In Japanese, orijinaru bideo animeeshon オリジナル ビデオ アニメーション is the broken English mashup that gave birth to the OVA acronym.
What is an OVA?Basically, disregarding the nonsensical words, an OVA is an anime that's not aired on TV. That is "originally" sold as video, that is, as a DVD or Blu-ray. This often happens for two reasons.
OVA Not Good for TV FormatMost anime airs on "TV", terebi テレビ, as "TV programs," bangumi 番組, and thus are called "TV anime," terebi anime テレビアニメ. However, in order for this to happen, modern TV anime must follow certain, rather strict guidelines.
For example, TV anime have episodes of 20 minutes, with rare exceptions. They air only once a week, with rare exceptions. And they air for one season or more. There are four seasons in a year. A year is 52 weeks. So every TV anime has to have 12 or 13 episodes, or 24, 25, or 52, 53, etc.
Most anime directors are skilled enough to handle these sort of restrictions pacing their story, stretching or skipping, placing cliffhangers accordingly and figuring out the best time to stop the adaptation. However, in some cases, airing the anime on TV would be too much of a hassle, and a more free format would be necessary.
This is most often the case with anime that don't have 12 episodes of material. They are original anime (not adaptations) and were written to fit perfectly into 3 episodes, or 5 episodes, etc. Since TV is not an option for them, they are sold directly as OVAs. That is, as Blu-rays and etc.
Example: Hellsing Ultimate (10 episodes of 50 minutes each), sold originally as DVDs.
OVA is Bonus
Another common case is when the OVA is a bonus. It's one or two full episodes or a number of shorts that come bundled with the purchase of a Blu-ray set or a certain manga volume.
In these cases, the OVA is going to be related to the story of the purchase. It might be a prequel, a sequel, a side-story, a spin-off, or even a satire. Some are really just bonus episodes, others are light-hearted slice-of-life-like episodes. It depends.
Example: Shingeki no Kyojin: Kuinaki Sentaku 進撃の巨人 悔いなき選択 (side-story / prequel about Levi)
What is an ONA?
Now that we know what an OVA is, it's that to know the ONA, OVA's successor!
With time, technology, and etc. things became internet-ey and anime was no different. The ONA, or Original Net Animation, which is often called "web anime," or uebu anime ウェブアニメ, follows the spirit of OVAs and puts anime in the web with online streaming instead of airing it on TV or selling it in DVDs and Blu-rays.
Literally nobody says orijinaru netto animeeshon オリジナルネットアニメーション, though. It's always referred to as the acronym ONA or Web Anime, because, seriously, look at how long those words are. Nobody's going to say that.
One ONA example is Bounen no Xandou 忘年のザムド, which has a format fit perfectly for airing on TV (26 episodes, 20 minutes each), but instead of going on TV it was published on the Playstation Store. Yes. On Playstation. The game console. You could watch this thing through the internet, so it was a web anime.
Later on, however, that ONA was sold as a Blu-ray. As if it were an OVA. But since it was originally published through the net, it is an ONA, not an OVA.
Likewise, if an anime is streamed in a service like Crunchyroll, Furnimation, Hulu, or even Amazon, but it was originally aired on TV it doesn't become an ONA. It's just a TV anime being streamed online. To be an ONA it has to be originally made to be streamed online. That's where the "original" word came from in OVA and ONA.