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every lil specific type of law, theres a lawyer for it!!!!! DUI, Criminal.. everything. specifics adn general
I'll try to answer, because I'm experienced at this stuff. I'm going to really simplify it too, because if I don't I'll end up writing a whole book. The first two are maximum payout amounts and the second two are what can happen that they'll cover.
This stuff is all a part of the Commercial General Liability package on your policy, and liability is when you're found legally responsible for someone else's loss or damage.
The aggregate is how much they'll pay for all claims within a stated term, usually the 1-year contract itself. It means the most they'll cover is two million dollar lawsuits, or one two million lawsuit, or four 500K lawsuits... you get the idea. They won't cover more than $2 million in liability during the year. NOTE: That's the amount paid out, not the amount you're sued for and legal expenses are covered in addition to that. They'll hire and pay a lawyer to defend you.
The 'each occurrence' means exactly that: They'll cover up to $1 million for each single thing you get sued for. So if something happens and 2 people sue you over it, each for a million, the most your policy will cover is $500K for each of them because it's one occurrence.
Products is any product you sell and are legally responsible for. Anything you package yourself and sell under your own label (like fresh meat) would be a perfect example. Completed operations would be more like a contractor who installs a sign and the sign falls on someone. Remember, I'm seriously over-simplifying this. There are a million qualifications, conditions and exclusions to all this stuff.
Damage to premises liability would be if you negligently damaged the premises you're renting (if you're renting), including buildings, outbuildings, the lot, etc. It'a not the same as Premises Liability, which is when you're held responsible for an injury because you own or control the premises it happened on or in. Damage-to-premises is also called Tenant's Legal Liability.
Again, I'm using seriously layman's terms. Don't take this answer as a full explanation, whatever you do.