Renters insurance covers a lot. It covers your stuff wherever it is, it covers what you're responsible for since you're living in someone else's place, and it covers the fact that you're counting on your place every day as a shelter.
You should know and build up your own emergency fund. The more you choose to cover on your own, the cheaper your policy. That's what your deductible is: what you cover on your own before your insurance would kick-in. It doesn't mean you pay out of pocket to make a claim, just that you'll split the costs with the insurer and they'll take that much out of your claim check when the time comes.
Renters insurance includes property coverage for when your belongings are lost, destroyed, or damaged. Your property or items–known as stuff and things you own in plain English–is covered no matter where it is. Your laptop and bag in a coffee shop. Your luggage on a trip. Your bike locked-up outside your office.
Most policies write out how much they'd be able to help in a total loss, losses that have lots of different kinds of property, and for every item lost. Usually, since you can't carry everything you own on your back, they'll cover less than your total property coverage when you're traveling or it's outside your place. Most people don't know or underestimate how much have they own. Insurance companies will ask you how much property you want to cover. Think furniture, electronics, jewelry, clothes, candles, plants, bikes, and more.
It'll cover you from fire, theft, vandalism, lightning, riots, and most natural disasters. Fraud and negligence aren't covered. Earthquake isn't covered but can sometimes be added for some extra money with a special endorsement. Flood isn't typically covered because they're so common, can hit so many people at once, and for so much damage that it becomes unaffordable or unfeasible to cover. Sometimes a company will help you buy flood coverage backed by the government–it can take up to 30 days to take effect so think ahead if you're in a possible flood area.
Living in someone else's place means you're responsible for what happens there. Liability covers other's people stuff that you're responsible for that gets lost, damaged, or destroyed. It doesn't cover your belongings. This is something landlords and property managers will ask you to get. It's very normal for them to ask you to have them added as an additional interest. It's not normal for them to ask to be named as an additional insured, which would allow them to make claims in your name. This is a red flag for other predatory practices–so beware.
Think about pets. Pets can damage other people's things or your apartment's walls. Ask if special coverage is needed for pets in your policy. Lawsuits aren’t cheap–$35,000 is common if your dog bites someone. A guest could get hurt in your home and ask you to cover medical bill or there could be a kitchen fire that damages your landlord's cabinets and he asks you to pay for replacements. Liability coverage could help you pay for that.
It will also pay for a place for you to stay, like a hotel, if your apartment becomes unlivable. There's usually a maximum amount for a day and for a single event. Still this is a crucial thing to help you bound back if a tree crashes into your place and there's a huge whole where your roof used to be.
Your landlord would be responsible for patching up the actual damage and would either stop asking you to pay for rent or you could ask for loss of use help–you can't get help from both places.