If a creature dying at the same time as Teysa (including Teysa itself dying) causes a triggered ability of a permanent you control to trigger, that ability triggers an additional time.

The trigger event doesn't have to specifically refer to “creatures.” In these cases, the trigger event may also refer to something being “put into a graveyard from the battlefield.” For example, an ability that triggers “whenever an artifact is put into a graveyard from the battlefield” would trigger twice if an artifact creature dies while Teysa Karlov is on the battlefield.

Look at each creature as it exists on the battlefield, taking into account continuous effects, to determine whether any triggered abilities will trigger multiple times. For example, if a land that has become a creature dies, an ability that triggers when it dies triggers twice.

Teysa affects a creature's own “when this creature dies” triggered abilities as well as other triggered abilities that trigger when that creature dies. Such triggered abilities start with “when” or “whenever.”

An ability that triggers when a creature “leaves the battlefield” will trigger twice if that creature leaves the battlefield by dying.

If a creature dying at the same time that another permanent you control leaves the battlefield causes a triggered ability of that permanent to trigger, that ability triggers an additional time.

An ability that triggers on an event that causes a creature to die doesn't trigger twice. For example, an ability that triggers “whenever you sacrifice a creature” triggers only once.

Teysa's effect doesn't copy the triggered ability; it just causes the ability to trigger twice. Any choices made as you put the ability onto the stack, such as modes and targets, are made separately for each instance of the ability. Any choices made on resolution, such as whether to pay a cost for that triggered ability, are also made separately.

If you somehow control two Teysas, a creature dying causes abilities to trigger three times, not four. A third Teysa causes abilities to trigger four times, a fourth causes abilities to trigger five times, and so on. This also means that if you control Teysa and cast a second one, an ability that triggers when it dies due to the “legend rule” triggers three times.