There were 4 main groups of traditional hunting methods.
Spears
Both thrown spears and thrusting spears were used. Sometimes, the spears were poisoned. Spears could have iron heads, stone heads, bone heads, ivory heads, or be simple pointy sticks. Generally, from the introduction of iron onwards, iron heads were preferred, with the other types having a long history stretching back into the Paleolithic when they were also used for other species, and outside Africa (yes, they hunted mammoths with pointy sticks).
A wide variety of tactics were used. A large group of hunters could surround the elephant, perhaps shouting to confuse or scare the elephant, and multiple spears thrown at it by the hunters. One or a few hunters, perhaps also using dogs or fire, could chase the elephant into an ambush, where spears could be thrown at it at close range. A daring hunter could attempt to sneak under the elephant and stab it from below with a spear. A hunter could wait in a tree above an elephant path, and spear the elephant from above as it passes under the tree. A single poisoned spear could be thrown, and then the elephant tracked until it succumbed to the poison.
This is, of course, risky, and hunters would sometimes be killed. The use of throwing spears allows the risk to be reduced somewhat, but it's still dangerous.
Judging by the Paleolithic evidence, the spear was the preferred technique at the time, and it appears to have been the preferred technique in more recent times. (The spear might be the method most likely to leave skeletal evidence, so the Paleolithic evidence might be biased, but it also appears to be the modern pattern.)
Traps
Traps can be used. The most common type is the pitfall. Stakes can be used in the pit to kill the elephant, or the hunter might kill it with spears once it is in the pit.
The other main type of trap is some heavy object (e.g., a log) with one or more poisoned sharp spikes/spears.
Natural features, such as swamps and cliffs, could be used.
Dogs and fire could be used to help drive elephants into traps.
Arrows
High draw weight bows (well in excess of 100lb) were used in East Africa to hunt elephants. Otherwise, less powerful bows were used with poisoned arrows.
Axes
This is a rare technique, and demanding on the skill and daring of the hunter. The hunter either needs to wait for an elephant to pass, or carefully stalk and elephant, and use the axe to cut the elephant's ankle tendons. Supposedly, sometimes an unarmed accomplice would deliberately distract the elephant to allow the hunter to get close enough to the elephant.
For a review of prehistoric and recent traditional elephant hunting techniques, see