Prerequisites -
Only use braid with a clean tip. You'll find that even though the actual solder collected on the braid goes back 1/4", the braid itself is stiff and hard at least a full inch back (relatively speaking, depending on braid size and solder amount). That "hard" area on the braid, even though clean and clear of actual solder, doesn't work well. You need the braid to be clean and
flexible on the end. If not, cut it off with the same cutters you use to cut leads (preferably a good pair of flush cutters).
The Act of Removing Solder w/Braid -
Place the tip of the braid and some extra directly to the joint you want to un-solder. Hold it there with the hot iron. Use enough braid so that the tip of the soldering iron is completely in contact with the braid, not the soldered joint. You're trying to make a sandwich, with the braid as the "filling." The soldering iron tip should only be heating the solder joint through the braid, but all three things (solder joint, braid, and solder iron tip) should be in close contact and "squeezed" together with pressure on the iron. Depending on tip size, this might be 1/8" to 1/4" of an inch (assuming you're soldering audio electronics and not kitchen piping). IOW, don't just stick braid on the joint and poke the tip of the iron onto it. Use the sides of the entire front of the tip - the part that collects solder and is tinned - to make contact with the braid on top of the solder joint. As for the braid, the very tip of the braid should also be in contact with the solder joint and the tip of the iron.
Think of Bounty paper towel commercials. The very tip of the paper towel, dipped into a liquid, will wick into the paper towel. In the case of de-soldering braid, you need the loose fibers at the tip to get the wicking started. Or at least, it's easier to get it started that way. Unlike the paper towel, however, you need the heat of the iron to turn the solder into liquid before it will wick.
Heat the braid and solder joint with the soldering iron tip (the whole tinned tip) completely in contact with the braid. You should see the braid eventually start to smoke from melting flux, then the solder should start wicking onto the braid. If this doesn't work, a great secret to using braid is to dip the tip of the braid into solder flux paste before doing the above. The heated flux will melt and flow/wick into the braid. At the same time, when the solder melts, it will also wick and flow into the braid. The extra flux will make all of this happen much more quickly, even though most braid (all?) is already impregnated with flux. As with solder, sometimes this isn't enough and you need to add more.
The downside is that the extra flux will make a big mess. Thankfully, every bit of it can be dissolved and cleaned up later on.
PCB Pads (holes) -
PCB holes require a special technique. Very often you'll completely remove the solder, but the actual hole is still filled. This is most often an illusion. Solder (in my experience) has a pretty high surface tension - just like water. Most of the time, that solder left in the PCB hole is simply the very thinnest of a bubble spanning the hole. In this case, you don't follow the advice given above. Instead, go ahead and place the braid across the hole, then poke the very tip of the soldering iron into the hole, slightly pushing the braid into it. The heat is usually enough to wick the last bit of solder into the braid. There may be so little solder in the actual hole that you may not even see any solder on the braid when done, but the hole will be clear. Always be careful. Poking the solder tip directly into a PCB hole is also a good way to remove the trace around the hole completely. If that happens, all is not lost, however - as you'll read later on.
Time Loops and Dental Picks are for More than Teeth -
There are instances where even the above doesn't work with removing solder from a PCB pad (hole). Don't be afraid to apply more solder in these cases. Once there's enough solder build up on the hole, the wicking action from the braid may remove all of it cleanly. Just be sure that when applying solder, you melt everything sufficiently that the added solder is fully combined with the solder remaining in the hole. If not, you may just be caught in a time loop of adding solder on top and removing it - never touching the actual solder in the hole. (Ask me how I know this!!) Even so, this may take several tries or more. I find a good dental pick is indispensable in these situations. It can simply knock the melted solder out of the hole.
A dental pick is also indispensable in pushing the part leads out of the holes, too.
The only reason I will use solder suckers are those cases when there is so much solder that the amount of braid gets filled up beyond reason. When you're using inches of braid at a time, you're better off with a sucker rather than wasting all your money on using up braid. Go back later to do the final cleanup with braid.
Cleanup -
It can all be a very dirty experience. You'll end up with a lot of flux and burned flux on the PCB, depending on how much solder you have to remove. Alcohol rinses and an old toothbrush should be able to remove all of the flux, though. If you're careful and make certain you have your heat adjusted just high enough to keep from actually burning the PCB, there's no reason you can't make the entire PCB look brand new and unused. I've done it many times.
Repairs -
If and when you do damage a PCB, remember - there's almost nothing that can't be repaired. All those cutoff resistor leads are good for something. Use them to span torn traces or insert them into PCB holes that have had their traces completely burned out. Leave enough sticking out to provide a soldering surface. You may need to do a few acrobatics if you're soldering multiple leads to the same broken joint. Solder-resistant clamps, hemostats, etc., along with a selection of different thickness pieces of wood to block up the PCB can help.
EDIT: FYI, I only use Kester rosin flux paste (looks like dirty yellow soap) and Chemtronics rosin braid. There are other braids that work well, but some may use nasty fluxes to impregnate the braid. Always look for "rosin." Also be sure to try to get braid widths that match your joints and soldering iron tip. That said, I only use 0.1" for everything, including SMD. Sometimes I wish it was thicker, but then I have the solder suckers when needed.
This is a great guide:
https://www.chemtronics.com/Content/Images/uploaded/documents/SIZINGGUIDE_LR.pdf
