Dear reader, you are lucky. If I had been writing on this topic not all that long ago I would have needed a small textbook to even begin an introduction to its arcane complexities. But the modern trend has been to simplify it all into a reasonable set of rules and principles that are - well, common sense based.
Put simply, the tenant has a much greater overall liability for injuries to visitors occurring on property owned by the landlord but rented by the tenant. Such injuries can take many forms:
(1) The postman is bitten by the family dog and comes down with rabies.
(2) A child chasing a runaway ball wanders onto the rented property and falls into a ditch, breaking his leg.
(3) A guest suffers an electric shock turning on a lamp due to defective wiring in a private home.
(4) A guest suffers smoke inhalation due to a fire in an office building that was caused by defective wiring.
The general standard used in most jurisdiction is the same "negligence, recklessness, and willfulness" standard that is used in most personal injury cases. The liability gets greater as you go up the scale from negligence to willfulness, including criminal liability in some cases.
Negligence is when you are not aware of the risk but should have been - you failed to check your home for necessary repairs, for example, and someone falls through a rotten stairwell. Recklessness is when you knew of the risk but ignored it - for example, you knew the stairs were rotten but delayed fixing them and then invited your mother-in-law over. Willfulness is when you intentionally expose someone to an unreasonable risk ("Let's put banana peels all over the basement stairs and film Linda when she falls - she's so CUTE when she's angry!") Note that in order for the offending conduct to be willful, you don't have to intend that anyone get injured - you just have to intend that they be exposed to an unreasonable risk of injury.
The foregoing reveals why tenants are generally more at risk than landlords - because they possess the property, they have more control over its condition than the landlord does and thus have more duties that can possibly be breached to result in an injury. The landlord's responsibility is at its greatest when he has a duty to keep certain aspect of the premises in good repair - for example, the landlord is usually responsible for complying with the fire code.