The fiercest droids that you'll be coming up against in Republic Commando are the massive spider droids, which appear only rarely, but are always a challenge to defeat. This is mostly due to their insane health and armor; it'll take you a good minute or two to kill them off, even if your whole squad is able to concentrate on them.
The spider droid doesn't possess any normal blaster attacks; instead, it uses its main cannon for single, powerful shots, as well as a pair of missile launchers for extra damage. Luckily, you can disable its ability to use the missiles by firing on the laser-sighting mechanism that pops up out of its back every once in a while. When you spot the red scanner that rotates around the droid, fire on its source to knock it out; this will prevent the droid from using missiles for the duration of the fight. Its primary cannon fire is still quite powerful, though, so you'll want to stay behind cover in most spider droid encounters and let your teammates do the work.
Luckily for you, each spider droid encounter will occur somewhere where there's multiple positions for your teammates to maneuver. The best maneuver for them will be anti-armor, but you'll often find grenadier or sniper positions around spider droids, as well, so don't hesitate to be diverse. When your squaddies are in position, all you normally have to do to finish the beast off is to just wait and watch your teammates fire away. They'll take damage, sure, but since they'll be behind cover, they'll usually survive long enough to take the baddie down, and even if one does get incapacitated, you'll usually have enough time to get to them, revive them, and get them back into position before the next soldier goes down.
That said, if you have one, a Wookiee rocket launcher will make short work of spider droids, so don't hesitate to use it if you have one.
Scavenger Droids
These lithe little enemies are the favored companions of Trandoshans; they're quite capable of killing things, and are also pretty handy at cleaning up their own mess. Their attack pattern isn't difficult to figure out; when they spot you, they'll shoot their laser beams at you as they rotate in the air around you. When hit, they'll rear back and start up a suicide run and attempt to impact you for extra damage. If they get close, they can also latch onto your helmet and drain your shields and health that way; use your fire button to wrench them off and throw them to the floor.
The main problem with scav droids is that they're wickedly difficult to hit, especially on the Xbox without auto-aim on. They do a really good job of avoiding incoming fire by bobbing and weaving; your best bet to hit them is usually to have a soldier in sniper position, if at all possible, as a single shot will blow them out of the sky.
Grievous Guards
Grievous' dread bodyguards are exceedingly rare enemies, but when they do appear, they're going to cause fairly big problems for you. They usually attack at close range with an electrically-charged baton, which can cut through your shields and take you down with just a couple of swings. Luckily, your teammates are more capable of resisting this damage, so you'll want to try and stay away from these guys and hope that they go after another member of your squad. They also pack homing missiles, similar in effect to the spider droid's missiles, which they'll fire after going to one knee and charging up. You'll want to be behind cover when this animation starts, if you suspect the missiles are coming your way.
Actually taking down a guard is a task in and of itself, thanks to the fact that they're highly mobile, acting more like Geonosian soldiers than your typical clanker, and due to their immense health reserves. Most guard fights take place in areas with multiple sniper and anti-armor positions, so you'll want to have your teammates set themselves up into these, with sniper positions probably being the most preferable option available. They're obviously not going to take much damage from blaster fire, so when you spot a guard, your best bet for dealing damage to it is to either unload anti-armor shells into it (assuming it's not directly next to one of your teammates) or by chucking EC grenades at its feet, assuming you can get it to stand still long enough for an accurate placement. This latter is probably the best method for draining their health, as you can use it while a guard is wailing on one of your teammates without him taking too much damage, and because it'll freeze the guard in place while your snipers get a few shots on it. It's normally easy for them to miss their shots due to the speed with which the guards can move.