S.H.I.V. Units – These are small tanks that can take the place of an XCOM soldier. They can't take cover, and don't have much more health than well armoured soldier, but they do have decent weapons and (if upgraded) can use the Suppressing Fire ability. Generally speaking a S.H.I.V is better than a rookie, but inferior to a veteran soldier. Their one saving grace is immunity to mind control, making them handy for taking out psychic aliens.
Skeleton/Archangel Armour – Both of these work the same way, they give their wearers the ability to reach higher ground, either by grappling hook or by flight. They are best used by Snipers, who benefit most from increased elevation.
Ghost Armour – While Archangel Armour's function is obvious, Ghost Armour is a little more uncertain. The grappling hook and light protection fits a Sniper, but the invisibility seems more suited to Assaults. In the end I preferred it on the latter, using the grappling hook to quickly close in on enemies and drop down on them from above, Spiderman style.
Grenades – Frag Grenades do three damage to all enemies in their radius, usually less damage than merely shooting someone, but their ability to fly over cover and destroy terrain makes up for that. They're a useful tool, and it's always worth taking a couple on mission. Autopsying a Cyberdisc will let you upgrade to Alien Grenades, which do five damage, and it's well worth doing.
Nano Vest/Chitin Plating – Both of these increase your health by a small amount, which useful in the early missions, but rapidly irrelevant when you have strong armour. You'll get more mileage out of Chitin Plating's secondary ability, which reducing incoming melee damage. This is a great way to protect yourself from Beserkers and Chryssalids.
S.C.O.P.E. - This offers an simple +10 bonus to aim, and also extra critical chance when upgraded. This is very useful to Snap Shot using Snipers, letting them overcome the penalty for moving and shooting. As your troops reach higher levels it becomes less useful, as their aim will be high enough already.
Medpacks – It's always worth taking a couple of Medpacks on a mission, they can restore hitpoints, cure poison and stabilise dying soldiers. Medpacks are best carried by Support soldiers, who can take several perks to enhance them. Be sure to autopsy a Thin Man as soon as possible in order to double the amount of damage you can heal.
Arc Thrower – The Arc Thrower is essential to the completion of the game, as it's the only way to stun and capture a live alien, but it's also a useful tool. It can easily take out low health enemies at short range, and the first captured alien of each type offers a research bonus. It declines in usefulness as aliens get more health, but becomes interesting again if you grab the upgrade that allows it to repair SHIV units.
Combat Drugs – Psychic powers getting you down? Why not snort the crushed bones of a Muton Beserker? Well for lots of reasons obviously, but there are positive effects too. Taking these Combat Drugs boosts your willpower and grants extra movement speed and damage resistance, making them useful for taking out psychic aliens.
Sectoids – Sectoids are weak and easy to kill, but like to scamper around and flank you. Their special 'mind link' ability allows one Sectoid to heal and buff another. If you spot this happening, aim for the Sectoid who started the link, this kills both enemies with one shot.
Thin Men – Thin Men are much like Sectoids, but quicker and more agile, able to quickly leap over obstacles. Their special ability is to spit a toxic cloud that poisons your soldiers. Be sure to avoid bunching your men up, or you'll quickly run out of medkits trying to cure them all.
Floaters – The unfortunately named Floaters are some of the more manoeuvrable enemies you'll face. Their flying abilities let them easily hop over obstacles, but can also often strand them out of cover. Occasionally they like to fly straight up in the air and drop down behind your men. Always have a man on overwatch to deal with this. Floaters will eventually be replaced by a souped up version called Heavy Floaters, which act the same but have greater health and damage.
Outsiders - Outsiders act as a sort of captain to enemy ships, having a chance to turn up in downed UFOs. They've got more health than the standard enemies you'll face around this time, and can do some nasty damage with their plasma rifles. Fortunately there's only ever one per ship, and they only appear once you enter the ship. This means the best strategy is to carefully sweep the area of all other aliens before venturing inside. One Outsider isn't a huge threat to a full squad, but triggering him in the middle of a large firefight can spell doom for inexperienced troops.
Chryssalids – Chryssalids were one of the original X-Com's most feared enemies, but they're marginally less terrifying this time around. While they retain their high speed, deadly melee and ability to turn their victims into zombies, their health is now firmly in the 'middleweight' range. Chryssalids will always rush straight at you (or a civilian) and try and engage in melee, which means that taking cover is less important than keeping your distance, and they'll almost always trigger overwatch fire.
Mutons – Mutons are essentially an extra durable ranged unit. Not as agile as Thin Men, but tougher and harder hitting. They also like to throw grenades at clusters of XCOM units to destroy their cover. Once a Muton reaches half health, it will use its Intimidate ability, triggering panic in units with low morale. It all adds up to a fearsome enemy, but there aren't really any special strategies to use beyond 'shoot them a lot'. Later in the game Muton Elites, tougher versions with larger guns, will start to appear.
Sectoid Commanders – Sectoid Commanders have more health and a stronger ranged attack than regular Sectoids, but usually eschew shooting in favour of mind controlling your men. This is very hard to deal with, especially as they often come in pairs and use Mindfray to lower the target's will before 'assuming direct control'. If this happens, kill the controlling Sectoid as quickly as possible to get your man back. The best counter to Sectoid Commanders are your SHIV units, which can't be mind controlled. Alternatively, strengthen your will with Combat Drugs or Psychic Inspiration.
Drones – Drones are fast, manoeuvrable flying enemies with low health and damage. They can be a real nuisance on Terror Missions, where they can fly around zapping civilians wily nilly. Most of the time however, they stick close to a Cyberdisc or Sectopod, using their repair abilities to restore its health. The best way to deal with a Drone is to hit the unit it's repairing with a Rocket or Alien Grenade, as the Drones will usually be taken out in the blast, while still dealing good damage to the primary target. Upgraded Arc Throwers can be used to take control of Drones, but since they're usually next to something scary, it's rarely worth it.
Cyberdiscs – The first real heavyweight unit you'll face, the Cyberdisc can fly in it's closed form, or open up to deal heavy damage and throw grenades a long distance. They are also often followed by Drones who will repair any damage it takes. The Cyberdisc's weakness is it's inability to take cover, making it easy to hit, and its vulnerability to HEAT rockets. It is advisable to tackle the Cyberdisc from as long as distance as possible, reducing it's chance to flank you or throw a grenade, and taking advantage of its erratic movement.
Muton Beserkers – The Beserker is a special kind of Muton, he carries no weapons, but has a fearsome melee attack and the ability to smash straight through cover. He also has another ability that is a little more interesting, every time he gets shot, he charges forwards a few steps. This can let him close in on your soldiers very quickly, meaning you are best off hitting him with a couple of heavy shots, rather than several light ones. Clever commanders can make this extra movement work to their advantage, hitting him from distance with a sniper in order to bring him within range of an assault, or forcing him to run in circles by shooting him with several different soldiers in different positions.
Sectopods – Sectopods are some of the toughest enemies you'll ever face. These walking tanks have truly enormous amounts of health, and can do tremendous amounts of damage. A Sectopod has three kinds of attack, a powerful direct beam weapon, an indirect artillery barrage and a plasma gun which is perpetually on overwatch, even if the Sectopod has fired this turn. Thankfully they are also large and slow, allowing you to avoid their fire arcs and slowly wear them down. Use grenades to attack them from a place of safety, burn up their overwatch shots with Lightning Reflexes, and only engage them head on if you have full cover.
Ethereal – The big daddy of the alien world. The Ethereal is the most fearsome opponent you'll ever face. Like Sectoid Commanders they love to mind control your men, but they also have a couple of other nasty tricks up their sleeves. The first is a psychic storm, which deals damage in a radius and lasts for several turns. For this reason you should avoid engaging an Ethereal in tight quarters. Secondly, they deflect any shots that miss them back at the unit that fired them, meaning you should only shoot when you've got a very good chance to hit. Thankfully the deflection doesn't work on area attacks, so feel free to spam them with as many rockets and grenades as possible.