Triune's encyclopedic list gives a good overview of what'll hurt you in the tactical squad battles of UFO: Enemy Unknown and how it'll do it. Unfortunately, it's a distant one about a game where battles get highly personal. This is one of those times where the enemies gain character just by being big enough bastards.
It's on this note that I choose to enlighten you about:
These grey, man-sized, clawed unholy whoresons are encountered en masse in Snakeman terror sites - where a large number of aliens descend on an urban area to blow shit up - and crashed terror ships, and serve to prove the mission names' aptness to the ill-equipped player. A Chryssalid's hand-to-hand attack irreversibly converts any human into a twisted hostile zombie that a new Chryssalid erupts from (silently, in an enviroment where sound is very important) a few turns later or on destruction. Against heavy weapons platforms, it's either deadly or an instakill.
The things move and attack at speeds that would merit "WOOSH!" sound effects, so distance is a poor defense, good luck getting a reaction shot in and if the first five strikes somehow fail they can usually squeeze in a couple more. They're tough, they've got superhuman night vision, with just one conversion the survivors will have to deal with three enemies instead of one to avoid being overrun, and they grin all the time.
A terror site means civilian presence, which means that Chryssalids get to enjoy target-rich enviroments while the X-COM soldiers sent to take them out are supposed to exercise restraint. And they do enjoy them, so every civilian is a potential Chryssalid. In a game already full of precious yet fragile soldiers, dark corners, twisting alleys and night-time battles this is just plain nasty. As the grisly sounds of zombification come from the shadows as the creatures find another survivor, and another, and another, or as the second rocket misses and two Chryssalids swiftly become four, these are the aliens that make you want to get out and nuke the place from orbit.
What are the countermeasures? First, moving in groups of three or four, with some space and enough weapons to take out the Chryssalid-Zombie-Chryssalid combo in one turn without any one member. Second, electro-flares are vital by night. Third, excessive firepower. Do as the Romans did to Carthage, only instead of sowing the ground with salt use grenades. Fire explosive shells full-auto. If you have suspicions about an alleyway, remove it. Just wait a bit for the smoke to clear. Some deliberately shoot civilians, an option that's worth keeping on the table. Lastly, and definitely the most importantly, flying suits. Chryssalids can't fly! Enough powered armor for a full squad is a painfully costly Elerium investment, but the very first time of dangling tantalizingly out of their reach just before dropping grenades and opening fire with plasma rifles makes it fully worth it. Smart players check their list of one-liners beforehand, but the grin that scares small children will happen on its own.