When the aliens' kill-loss ratio is in their favor, it begins to affect your team's morale negatively, causing them to panic and suffer other morale failures. But as it turns out, the aliens are subjected to these rules too, and if you're winning, expect to see the message "Alien-name Job-name has Panicked/gone Berserk!"
"Triscene Terrorist" (8-foot-tall dinosaur with plasma cannons on its shoulders) has panicked.
Setting off a bomb in a gas station during a terror mission leads to a long chain of explosions. It can also be a great way to clear a bunch of aliens around said gas station. Just be careful of the huge smoke cloud it leaves.
This troper once found a bunch of Sectoids fueling up their Cyberdisc. A few laser shots and it went BOOM. Then the pumps went BOOM BOOM BOOM BOOM. Cue alien death shrieks in polyphonic stereo.
Sometimes, aliens will try to throw grenades or fire stun bombs at close range, which often leads to them killing or stunning themselves, respectively.
It's even worse than that. This troper saw an alien fire a stun bomb at an HWP (Basically a tank), which are immune to stun damage (and no, they didn't have the excuse of reaction fire as it happened during their turn).
Two of the alien races have the ability to use psi attacks on your troops, making them have morale failures or taking control of their minds in doing so. Once you've researched the technology to allow your soldiers to do the same, have fun assaulting the weak-minded alien Mooks.
Mind controlling an alien to attack his companions and watching as they subsequently go berserk, unloading everything they have into someone no less than 4 feet away, is quite a sight.
The Lobstermen in Terror from the Deep are nasty, but hearing their amusing death scream makes the effort of killing them worthwhile.
All aliens on a landing cleaned up besides 1, looters(no armor no weapons) deployed, the most worthless one of the bunch opens a door, scares the sectoid commander who fires off his stun bomb launcher, the bomb bounces off the soldier's non-existent armor, hits the sectoid in the head knocking him out, then explodes stunning the soldier and ending the mission, he got promoted for that, and we caught a live sectoid commander.