You can hit the monster on your own (solo) but if there is a party hunting it as well then you wont do enough overall damage and likely won't get credit for doing it.
Thus the reason for the invite.
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If you see a specially-named level 50 monster on the map you're on that has a blue (non-aggroable) symbol, it's a B rank. You should shout its name and position for anyone searching for it for their weekly, so that they can find it more easily.
If you see a specially-named level 50 monster with an orange (aggroable) symbol, shout its name and position too so that people can pass it along to their hunt linkshells. If it's a huge one, it's probably an S rank.
If you're unsure of what it is, you can check [this] page.
You can participate in A & S rank hunts, and you should receive allied seals (the hunt currency), but you will die every single time something hits you if you're below about 40. That's just an estimate...I've seen level 5-20s do it, and 45+, but never in between.
A and S ranks aren't weekly or daily or anything, you just do them when they spawn. You will never receive bills for them. Tbh, I don't know how Heavensward ones work...I'm only 56 right now.
More specifically, you're unlikely to get full credit solo. The minimum you can get is 4 Allied Seals and 3 Poetics tomes just for tagging an A rank as opposed to getting up to 40 seals and 30 tomes if you and your party do major damage to it. There are also some intermediate levels in between those for players who did more than just tag it but not quite enough for full credit. (For an S rank, full credit is 100 of each, so I expect the lowest tagging score would net around 10 each if they have the same 1/10 pattern as on A ranks.)