Bastard doors are doors that slow Link down from his usual 1.5px/frame speed to 1px/frame if diagonal inputs are used while inside the doorway. Obviously you'll want to use cardinal directions to avoid this slowdown, though it can seem counterintuitive at times.
You will notice that all of these doors face east/west, and all of them are through locked doors. You only have to worry about the direction you hold on the other side of one of these doors if it is a subtile transition on the underworld map. So for example, the door in Eastern Palace moving from Dark Switch room to the Catwalk Above Cannonballs is a locked east/west door, but it moves through a supertile transition and is therefore NOT a bastard door. Another important rule to know is that movingor out of east/west facing doors will always slow your movement unless it's a shutter door or bombwall. This page lists instances where certain diagonal movements will also slow Link's speed.
An additional quirk of these doors is that if you do get the slowdown, the game will allow you to use items inside the doorway. This saves time in one instance in NMG speedruns that I'll note below.
A WQ tutorial on door movement:
1st Popo Room Door
First bastard door in Desert Palace is this one going into the first popo room. Although it seems like you'd want to hold diagonal here to get to that first popo, you shouldn't as it will give you some slowdown in the doorway. Hold cardinaluntil you've cleared the doorway, and then go to to get to the popo.
Dark Fast Knights Door
This is the room right right after dark maze which has the two fast knights. You can opt to dash in this room, which is a little easier, but walking is faster if done correctly. If you do walk, you should be avoiding the bastard door here, which gives slowdown on. You'll need to hold here until clearing the doorway, and then go to diagonal.
Firesnakes & Zazaks
You can't holdwithout getting a speed reduction here. You have to hold until you've cleared the doorway before going to up and then dashing through the room.
Here you can't hold either diagonal (or ) without a speed reduction. Here I would actually advise taking the speed reduction, since you can get hit by the spike if you don't.
Mire Tile Room
You'd think you can just holdthrough this entire room, but that'll give a speed reduction in the doorway. Need to hold cardinal until you're out of the doorway, and then you can go to diagonal.
Spike Warp Room
Shouldn't holdout of this door, but instead use . This one is a little tough because with just holding right out of the door, it can be easy to run into the crystal switch, so be careful going for this bastard door avoidance.
Here is the instance of using a bastard door to get an early item use. By intentionally holding diagonal inside this doorway, you can plant the bomb early. This is the fastest method of doing the room.