The liability coverage triggers for anything from people getting hurt on the property to (other) people's stuff accidentally getting damaged on the property. Basically anything anyone could sue you to pay for if it occurs on the property.
The personal property coverage is what covers your stuff if anything happens to it.
A landlord's policy will generally cover two things as well: liability and structure (and possibly property, e.g. major appliances that are supplied with the unit).
Landlord's requiring that tenants have a renter's policy specifically require liability coverage (usually a minimum amount, such as $100k), and don't care if you have personal property coverage. By doing so, they ensure their own premiums stay low because liability claims will go against the tenant's policy first, and only fall back to the landlord if the liability claim goes over what the tenant's policy will cover. So if a tenant has a $100k liability limit, and someone wins a $120k liability claim, then $100k of that gets covered by the tenant's policy before the landlord's insurance even comes into play. If someone wins a $50k claim, then the landlord's insurance never has to deal with it.