The AI in Fire Emblem games tend to have very poor decision making skills. Archers will often go straight after mages, which are often the only units that can counterattack from that distance, and they can even do this if they need to be one square from your Lord to do so. Other enemies will see a line of five soldiers and will ALL choose to swarm either the one riding the dragon who kills things in one hit or the heavily armored one who doesn't take damage. And the bosses have this weird concept that the best way to fight the heroes is to stand perfectly still in their room until you're within range, even while the heroes start filing in around them. In most cases, unless they are scripted, the AI will most of the time choose to attack any units that gets into their attack range, even when they are going to do no damage on the target (like some cavalier with wimpy weapon attacking a knight, or a mage attacking a high-res bishop).
The whole suicidal attack on the mages thing is actually explainable. The AI is aiming to cause anyfatalities it can, no matter the cost to its own units. Keep in mind that winning a match in Fire Emblem is quite easy — winning with everyone alive is the rub. Given a 1% chance of an instant kill, and a 100% chance of doing half of the max hp of the target, the AI will go for the 1% instant kill chance every time and hope it gets lucky just to spite you. While most of the time you will laugh at the foolishness of the computer, when it finally manages to get that hit in...
To be more precise (at least in the GBA ones), the AI first targets any units it can deliver enough damage to kill in one hit, regardless of the likelihood to hit or the damage that it will receive, and if it can't kill anyone, then it goes for the one that it can deal the most damage to, again not counting likelihood to hit. The issue is that the people with the lowest defense are generally your magic users. They also happen to have ungodly dodge most of the time (at least for Sages), and are fully capable of one-shoting most units in the game once they've been trained. The GBA games do not take into account whether they'll be counter-attacked at all, leading to sending a Sage out in front of your army and letting him destroy half the units on the map being a legitimate strategy. And because of the way probability to hit works in this game, a 30% chance to hit is really more like a 10%, and that's the sort of hit rate opponents often get.
The stand still thing applies mostly to bosses who are on seize points, and if they were to move, then you could rush right up and seize the gate/throne.
In an early stage in Radiant Dawn, you are forced into using only two units: the Squishy Wizard Micaiah and the Black Knight. Since the enemies are there specifically to kill Micaiah and the Black Knight is there specifically to protect her, you'd think they'd send the entirety of their oddly large force at her at once. Nope, they go two or three at a time and hack at the first living thing they see.
Even more fun is that enemies will always, always go straight for a unit that has no weapon equipped, as they (obviously) can't counterattack (or if laguz, not very well). You can easily get Micaiah to level 20 without a problem on that map by parking the Black Knight somewhere with his sword unequipped and letting the enemies flail pointlessly at him while Micaiah whittles them down from afar.
This has applications beyond power-leveling. Many Fire Emblem veterans know that the best way to save a mission that's going pear-shaped is to unequip your strongest character's weapon. Picture this: the enemy has three swordmasters standing next to a mission-sensitive character, who has only a few HP left. In order to win the map, the enemy need only attack with a single unit. Yet if you move an armored unit up and unequip their weapon, any enemy unit within range will immediately abandon their attack on the almost-dead Lord and attack the armored unit instead, even if they can't damage it.
This is nicely averted with bosses who seem to stand still on the throne. So you carelessly move your Squishy Wizard up, planning to attack next round, when the boss runs right up and kills your exposed wimps. Oops.
Then again, this makes it possible to slip past them and seize the throne for an automatic win by sending a disposable unit into their attack range at low HP. On some maps, it can be worth spending a deployment slot on an untrained rookie to have them serve as bait in this way.
Which extends to neutral/allied units as well. Particularly noticeable in 3-10, where, for example, the leader of the Crimean Knights, Geoffrey, will move right up to a Bishop and then not attack it, but then Astrid will take a shot with her longbow. (Thankfully, the longbow is very inaccurate, as this was one of those situations described below where the enemy was being spared for thieving purposes.)
Neutral units fall into this, especially if they're the ones you have to talk to and recruit. For instance, when Erk shows up in the second story arc of Blazing Sword, he comes out of a village and polishes off two mounted units before you get to him. If you don't get to him immediately thereafter, though, he'll run off and provoke a boss with high resistance and physical attack.
They also have a tendency to attack enemies that you don't want attacked just yet (usually because they have an item that can be stolen), throw themselves at enemies far more powerful than they are, and always make their moves in the exact same order, which means that on the rare occasions that one of them does have some sense of self-preservation, they'll often retreat when it's really not necessary. And they always retreat when it's time to heal, even if they're using a healing item. Even if it means leaving a plot-sensitive character exposed.
The Radiant Dawn AI also prioritizes units who have "rescued" someone. In theory, this makes sense, since if you rescue someone, the rescuer loses half Skill and Speed, making them sitting ducks who can't hit the broad side of a barn. Except tanky units don't care (at least for survival), and more importantly, they still do that if the rescuer has the skill that nullifies the stat penalties, meaning Tibarn (who starts strong and has that skill) can easily make all enemies flock to him and kill them on counterattacks, just by grabbing someone. Who needs Provoke?
The AI also has a tendency to ignore your equipment. Meaning that flying units will attack Mages with wind magic and armored units will attack Swordmasters (who, like Mages, have low defense and high avoid) with Armourslayers.
Then you get the units who are so outclassed by every player unit in range that it's literally impossible for them to actually do any damage (because they have a 0% chance to hit or 0 expected damage — or both) but insist on attacking anyway.
Enemies also have absolutely no experience with such advanced concepts as "strategic retreat", "regrouping", or "mixed unit tactics" beyond, as stated above, going after the most vulnerable unit available. However, this is circumvented by good level design and the player's general mindset... Usually. Having a single little soldier charge blindly at you when his allies are still too far away to assist is an unfortunately common occurrence.
The AI in Path of Radiance seems to prioritize doing damage/lowering their own damage taken above actually killing your characters — for example, putting Nasir in range of Ashnard will cause him to attack Nasir even if he could kill Ike with his next hit.
Ashnard will also attack Nasir at melee range, even though his weapon can attack at range (and Nasir then can't counterattack).
In Awakening, NPC characters are even more brain-dead than usual for the series. One mission, involving the players escorting a group of villagers, will have said villagers running straight into the enemy's waiting swords when they would be safe if they just stayed put. The villagers in this mission are actually following a fixed route towards the bottom corner of the map (the prior cutscene establishes they're fleeing in a blind panic) and will escape if they get there, but since that's going right through a thicket of enemies and the mission is usually over before they get close, it makes them seem like lemmings.
The AI in Awakening also pays no attention to a player unit's pair-up partner. They will gladly rush towards that unarmed Troubadour, completely oblivious to the fact she'll be replaced by a General by the time they get there.