Check the AOS page on this.
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The first part of the name that we use for identification purposes is the genus (plural form genera). This should be capitalized. Technically that's true even if we're abbreviating.
Paphiopedilum, Oncidium, Cymbidium, Phalaenopsis
Paph., Onc., Cymb., Phal.
The second part of the name denotes the hybrid name or species name, if known. This part should be lowercase if and only if the plant is a species, otherwise this part should also be capitalized to indicate it is a hybrid.
Phalaenopsis amabilis = SPECIES
Brassavola Little Stars = HYBRID
Sometimes there are additional parts of the name, though.
The most common things you see are clonal names & orchid society awards. The clonal name is the name of a specific plant and divisions, keikis, or clones of it. You can give any seed grown plant your own clonal name as long as it doesn't already have one. Clonal names go in quotes, like Phalaenopsis schilleriana 'Pink Butterfly' AM/AOS. That last part, the AM/AOS part, is an award of merit from the AOS. Awarded plants should have a clonal name and you should list the award as well, as it indicates flower quality. Clonal names should be written like hybrid names, the first letter of the words are capitalized. Awards are to my knowledge always all caps.
Sometimes you'll also see variety and form used in names. These are indicated with lower case and the indicator (variety, var., v. or form., fma., f.) is often abbreviated.
Phalaenopsis bellina var. caerulea
Phragmipedium besseae var. flavum
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Something to consider is that you really don't have to spend much time thinking about what's right. It's extremely easy to google the name of your plant, and generally that's going to get you the correct capitalization on at least the genus & species. Look at what the nurseries are calling it, they generally care about all this stuff, and while they may not always be up on the latest nomenclature, they'll at least have the names capitalized and quoted correctly.
Another common problem that people have is they mention their plants by species or hybrid name only. Things like "Sanderiana In Full Bloom". In the right context, you can probably get away with this. For instance if you're having a conversation about Phals and you mention sanderiana, it's clear from context what you're talking about (though still preferable to include the genus). In a post title or absent context, it's a really bad practice to get into.