Pick My Unit Run
That one run that everyone does. In this challenge, you have people choose units for you to use throughout the entire game. The number of units that can be used is entirely up to you. If reclassing is possible, you may also have people pick your unit's class.
RNG Run
This is similar to the Pick My Unit run, but instead of having people pick your units for you, you will be using a randomizer to pick your units. Again, the number of units that may be used is entirely up to you.
Ironman Run
To put it simply, this is a run where you play on Classic mode (if there's an option for that) and do not reset the game at all. Since fallen units are permanently gone, this challenge may get interesting when units start dying. Whether you lose the challenge upon getting a Game Over or get another chance at playing is up to you.
Casualty Run
Before every chapter starts, you randomize one unit (or more) that you have recruited to be deliberately killed in the chapter. Obviously, Lords are an exception to this, since that would automatically lead to a Game Over. Other than that, anyone may get a chance at dying, even if they recruit another unit later on.
One Weapon Type Run
A run where you try to beat the game using only one weapon type. The units you use don't matter as long as you stick to one weapon type.
One Gender Run
You may only use units of a gender of your choice. It would be better to choose the females, though, because there'd be less units to work with.
One Affinity Run
You may only use units with the same type of affinity.
Low Turn Count Run
A challenge where you try to play the game in the lowest number of turns possible.
Low Level Run
A challenge where you try to play the game with the lowest unit levels possible.
Solo Run
Your goal here is to try and beat the game with only one unit of your choice being allowed to battle with weapons.
There's a
general thread dedicated to Draug Solo Runs for FE11 made by
Bricktoad.
0% Growths Run
All of the playable units of the game have 0% growth rates for everything. In other words, you won't gain any stats from leveling up and you're forced to rely on units with high base stats and stat boosting items. You'll need a ROM and a patch for this challenge. (Look it up yourself if you want to try this.)
Reverse Recruitment Run
The playable units can be recruited in reverse order (Example: In FE6, Roy will be replaced with Karel). The unit's base stats, starting items, and class are also modified depending on when they're recruited in the game. You'll need a ROM and a patch for this challenge.
Randomized Recruitment Run
Same as the reverse recruitment run, except the order of recruiting units is in a randomized order.
Randomized Classes Run
A challenge where the playable units have randomized classes. Their starting items and sprites have also been modified. Commonly known as simply a randomized run. You'll need a ROM and some randomization program for this.
Randomized Weapons Run
The weapon properties (Mt, range, uses, etc.) are completely random. Enjoy playing with 3-10 range 10% crit 18 Mt Iron Axes! You'll need a ROM and some randomization program for this.
Chain Play/Hot Potato
While this may not be considered a "challenge" sometimes and can be attempted with pretty much any game and not Fire Emblem specifically, it's still pretty fun to do since it involves multiple people. Each person participating in this run will play a portion of the game and then send the save file over to the next person. They keep doing this until they beat the game. You aren't usually bound by self-imposed rules or restrictions, unless you're doing a combination of this challenge and some random challenge run.