In the natlang world, a trigger language has Austronesian alignment and therefore often a richer set of voices than just the more familiar active/passive dichotomy. Due to an apparent misunderstanding about the natlang situation, in the conlang world, a trigger language can also refer to a non-natlang system for giving one noun phrase prominence by displacing its case marker.

In Natlangs

Under Austronesian alignment, among the various noun cases are two, the direct case (ᴅɪʀ) and the indirect case (ɴᴅɪʀ), whose content semantics depend on the verb's voice and whose only invariant distinction is that the single noun phrase in the direct case is more prominent than any in the indirect case. The direct-case noun is said to have Austronesian focus, which is rather badly named, since it is has nothing to do with grammatical focus, a different prominence specifically indicating new information or even contradiction. Confusion of Austronesian focus with topic/theme, as in topic-comment languages, is also perennial in conlanging discussions. I feel like the only reasonable one out of the common analogies is that "Austronesian focus = subject" because this gives the correct understanding of verbs in the equivalents of the active and passive voices:

Tagalog
Kumain ng isda ang pusa sa sahig.
k<um>ain            [ng    isda]  [ang  pusa]  [sa   sahig]
<ᴀᴄᴛꜰᴏᴄ>eat[ᴄᴏᴍᴘʟ]  [ɴᴅɪʀ  fish]  [ᴅɪʀ  cat]   [ᴏʙʟ  floor]
Ate the fish the cat did—on the floor.
The cat ate—on the floor—the fish.

Here the verb has an action-focus (ᴀᴄᴛꜰᴏᴄ) infix and hence the action-focus voice, which behaves like other languages' active voice: the Austronesian focus or quasisubject "cat" is the agent in the direct case, the patient "fish" appears instead in the indirect case, and the location "floor" appears in the oblique case (ᴏʙʟ).

Tagalog
Kinain ng pusa ang isda sa sahig.
k<in>ain            [ng    pusa]  [ang  isda]  [sa   sahig]
<ᴏʙᴊꜰᴏᴄ>eat[ᴄᴏᴍᴘʟ]  [ɴᴅɪʀ  cat]   [ᴅɪʀ  fish]  [ᴏʙʟ  floor]
Eaten by the cat the fish was—on the floor.
The fish was eaten—on the floor—by the cat.

Whereas here the verb has an object-focus (ᴏʙᴊꜰᴏᴄ) infix and hence the object-focus voice, which behaves like other languages' passive voice (minus the need for any periphrasis to specify the agent, as, say, with English "by"): the Austronesian focus or quasisubject "fish" is the patient in the direct case, the agent "cat" appears instead in the indirect case, and the location "floor" still appears in the oblique case.

The big divergence from active/passive systems is the possibility of other voices that bring to prominence—again without needing periphrasis—nouns in other roles. For example:

Tagalog
Kinainan ng pusa ng isda ang sahig.
k<in>ain<an>        [ng    pusa]  [ng    isda]  [ang  sahig]
<ʟᴏᴄꜰᴏᴄ>eat[ᴄᴏᴍᴘʟ]  [ɴᴅɪʀ  cat]   [ɴᴅɪʀ  fish]  [ᴅɪʀ  floor]
Where the eating was—of the cat, of the fish—was the floor.
The floor was where the cat ate the fish.

Now the verb has a locative-focus (ʟᴏᴄꜰᴏᴄ) infix-suffix combo and hence the locative-focus voice: the Austronesian focus or quasisubject "floor" is the location in the direct case, and the agent and patient are both demoted to the indirect case, having what is conceptually almost a genitive-like relation to the action. (In Tagalog, for instance, the indirect case marker and the genitive marker coincide.) This large flexibility is why many linguists resist calling the Austronesian focus by the narrower-seeming term "subject".

Depending on the language, there may be many such voices that can turn noun phrases from a wide range of roles into a quasisubject simply according to a marking on the verb. There may also be other semantic side-effects, like Tagalog's tendency to treat direct-case nouns as definite, but this is the core idea.

What about the word "trigger"? It is a term used variously, by some as a synonym for the Austronesian focus, by some for the voice, and by some for the exponent of the voice, but the underlying commonality seems to be the observation that a marker change in one part of the clause "triggers" corresponding re-marking elsewhere.

In Conlangs

While many conlangers are aware of the meaning described above, there also seems to be a competing second meaning, which is an inequivalent system inspired by natlang trigger systems or perhaps a misunderstanding of them. Under this kind of trigger system, there is a trigger case (ᴛʀɢ) which is somewhat like the Austronesian direct case in that it conveys prominence and a quasisubject role. There are two differences, however:

1. The noun phrase put into the trigger case has its semantic case migrated to the verb, where it fills in for the exponent of voice.
2. Other noun phrases' roles are still always distinguishable from their case marker alone.

Let's compare, even keeping the word order from the Tagalog samples:

eat-ᴘʀꜰ-ᴀ  fish-ᴘ  cat-ᴛʀɢ  floor-ʟᴏᴄ
Ate the fish the cat did—on the floor.
The cat ate—on the floor—the fish.

Unlike the Tagalog example, there is no distinct ᴀᴄᴛꜰᴏᴄ exponent, and the migrated agent marker (ᴀ) serves that purpose instead. (One might consider glossing it as ᴀᴄᴛꜰᴏᴄ anyway since ᴀ on a verb needs explanation, but here I wanted to show the migration.) Moreover, fish-ᴘ is recognizable as a patient (ᴘ) even out of context because it uses a dedicated patient case, not a multipurpose indirect case.

eat-ᴘʀꜰ-ᴘ  cat-ᴀ  fish-ᴛʀɢ  floor-ʟᴏᴄ
Eaten by the cat the fish was—on the floor.
The fish was eaten—on the floor—by the cat.

Unlike the Tagalog example, there is no distinct ᴏʙᴊꜰᴏᴄ exponent, and the migrated patient marker serves that purpose instead. Moreover, cat-ᴀ is recognizable as an agent even out of context because it uses a dedicated agent case, not a multipurpose indirect case.

eat-ᴘʀꜰ-ʟᴏᴄ  cat-ᴀ  fish-ᴘ  floor-ᴛʀɢ
Where the eating was—by the cat, of the fish—was the floor.
The floor was where the cat ate the fish.

Unlike the Tagalog example, there is no distinct ʟᴏᴄꜰᴏᴄ exponent, and the migrated locative (ʟᴏᴄ) marker serves that purpose instead. Moreover, cat-ᴀ and fish-ᴘ are still in distinguishable cases, not lumped together in an indirect case.

As you can see, this has a similar feel, but differs meaningfully in details. It is also not quite naturalistic, and it is not known to appear in any natlangs.