The landlord's insurance covers the building, not your things. Renters insurance covers you. Many leases require it, and the landlord asks for proof before you get the keys.
Your belongings (personal property): if they are stolen or damaged by fire, smoke, a burst pipe, and some other events. Often also outside home, for example a laptop stolen from your car.
Liability: if you accidentally hurt someone or damage someone else's property, for example your bathtub overflows and damages the apartment below.
Loss of use (additional living expenses): hotel and food costs if your home cannot be lived in after a covered event like a fire.
What it usually does not cover
Floods from outside (rising water from rain or rivers). Flood insurance is separate, including contents-only flood policies for renters through the National Flood Insurance Program.
Earthquakes, usually a separate policy.
Your roommate's things, unless they are on the policy.
Very valuable items (jewelry, art) above a small limit, unless you add them.
Pest damage and normal wear.
Premium: what you pay, monthly or yearly.
Deductible: what you pay yourself before insurance pays. With a $500 deductible and a $1,200 stolen laptop, insurance pays $700.
Replacement cost: pays to buy a new item. Actual cash value: pays what the old item was worth after age. Replacement cost costs a little more and pays much more.
Renters insurance is usually cheap compared with other insurance. For many people it is in the range of about $10 to $30 a month, depending on the state, the amount of coverage, the deductible and your claims history. You can usually get it without an SSN, though some companies ask for one.
Make a quick list of what you own and its value (phone video of each room helps).
Get quotes from two or three companies online or from your car insurance company (combining can be cheaper).
Choose personal property coverage, liability (often $100,000 is the minimum the lease wants), and the deductible.
If the lease requires it, add the landlord as an interested party so they get notice if the policy ends.
Send the declarations page (the summary page) to the office.
Buying the landlord's own plan without comparing. Some only protect the landlord.
Letting the policy lapse when the card on file expires.
Not making a list of belongings, then not remembering what was lost.