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.”

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.

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.

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

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.

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.

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.

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.

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.

An ability of a permanent that triggers when a card is put into a graveyard “from anywhere” triggers twice only if Teysa and that permanent are both still on the battlefield immediately after the creature has died.